UNIVERSITA DI PISA Facolt a di Scienze Matematiche ... · UNIVERSITA DI PISA Facolt a di Scienze Matematiche, Fisiche e Naturali Corso di Laurea in Matematica Tesi di Laurea Magistrale

UNIVERSITA DI PISA

Facolta di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea in Matematica

Tesi di Laurea Magistrale

C∗-algebras and B-names for Complex Numbers

Relatore:Prof. Matteo Viale

Corelatore:Prof. Mauro Di Nasso

Controrelatore:Prof. Roberto Dvornicich

Candidato:Andrea Vaccaro

A.A. 2014-2015

A Thias e Sole

v

Contents

Introduction vii

1 Functional Analysis 11.1 Banach Spaces and Weak Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 C∗-algebras and Gelfand Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 First order logic and Boolean Valued Models 72.1 First order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Boolean Valued Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Set Theory and Forcing 333.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Boolean Valued Models for Set Theory . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Generic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 More on Cohen’s Forcing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Absoluteness results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 C∗-algebras and B-names for complex numbers 494.1 A boolean valued extension of C . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 C∗-algebras as B-valued models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 B-names for complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

vii

Introduction

In this dissertation we will present some connections between the theory of commutative unitalC∗-algebras, a specific domain of functional analysis, and the theory of Boolean valued models,which pertains to logic and set theory. More specifically, the main purpose will be to show thata commutative unital C∗-algebra A, whose spectrum is extremely disconnected, can be identifiedwith the B-names for complex numbers in the boolean valued model for set theory V B, where Bis the complete boolean algebra given by clopen sets on the spectrum of A.

This study is divided in four chapters. Chapters 1, 2, 3 provide the tools needed in Chapter4.

Chapter 1 is devoted to functional analysis, here the basics of the theory of C∗-algebras areoutlined. We are interested in commutative unital C∗-algebras. These, by Gelfand-Naimark’sTheorem, are Banach spaces of the form C(X,C), with X a compact Hausdorff topological space.

Chapter 2 introduces the logical tools we will need in chapters 3 and 4. Sections 2.1 and2.2 are an overview of elementary model theory and of the basic properties of boolean algebras,respectively. In the last section of Chapter 2 we introduce the notion of boolean valued model foran arbitrary first order language L, and we study the theory of these structures from a generalpoint of view. Given B a complete boolean algebra, the concept of B-valued model M comesfrom pure set theory, and generalizes the usual two-valued Tarski semantics, associating to eachformula ϕ a value JϕKM ∈ B. Boolean valued models are generally used to obtain independenceproofs, and they are strictly related to the forcing method introduced by Cohen to prove theindependence of the continuum hypothesis.

In chapter 3 we present some classical results from pure set theory and we briefly introduceforcing. The forcing method, developed by Cohen in 1963, is the most powerful tool (as of now)used in mathematics in order to prove independence results. However in section 3.5 we reversethe common perception of forcing, and we show how to use forcing as a tool to derive theoremswithin ZFC by means of the concept of generic absoluteness.

Finally, in Chapter 4, we build a bridge between the theories exposed earlier. In section 4.1 weintroduce and study the space of functions C(St(B), X) where B is a complete boolean algebra,St(B) is the space of ultrafilters–or equivalently maximal ideals–on B, and X a topological spacewith some specific properties. In particular we show that this type of spaces can be interpretedas B-valued models which contain X as a proper submodel. In section 4.3 we analyze CB, theset of B-names for complex numbers in the boolean model V B. This family of objects is shownto be isomorphic (in the sense of B-valued models) to the space of function C+(St(B),C), whichis the set of all continuous functions from the Stone space of B with image in C ∪ ∞ (seen asthe one point compactification of C) such that the preimage of ∞ is meager.

The boolean isomorphism between CB and C+(St(B),C) might be an interesting tool totranslate ideas and results arising in set theory to ideas and results arising in the study ofcommutative C∗-algebras, and conversely. This could be the case since, by the results of section4.2, commutative C∗-algebras can be studied in this context appealing to Gelfand Transform:

viii

given a commutative unital C∗-algebra A with extremely disconnected spectrum, there is anisomorphism (which can be defined using the Gelfand Transform) of the C∗-algebras A andC(St(B),C), where B is the boolean algebra given by clopen sets in the weak∗ topology on thespectrum of A. By means of this isomorphism A can be therefore embedded in CB, and thespectrum of A is mapped homeomorphically on the Stone space given by the ultrafilters (ordually by the maximal ideals) of the boolean algebra B.

We conclude the fourth chapter with an application of the absoluteness properties proved inChapter 3. In particular, we will prove that Σ2-formulae are absolute between the first orderstructures C (even endowed with arbitrary Borel predicates) and CB/G ∼= C+(St(B),C)/G (wherethe latter is also endowed with natural liftings of the Borel predicates), which means that thetruth value of these formulae is the same in C and C+(St(B),C)/G. We remark that in thiscontext C+(St(B),C)/G is the ring of germs of continuos functions f : St(B) → C in the pointG ∈ St(B).

A concrete example is given letting B = MALG, the complete boolean algebra of measur-able sets modulo null measure sets in C: in this case we obtain by Gelfand’s transform thatC(St(MALG)) ∼= L∞(C). We can now consider L∞+(C) to be the family of Lebesgue measurablefunctions with range in C ∪ ∞ which takes value ∞ on a null set, and lift the above isomor-phism to an isomorphism of C+(St(MALG)) ∼= L∞+(C). We conclude that whenever G is anultrafilter on MALG, L∞+(C)/G is an algebraically closed field extension of C which preservesthe truth value of Σ2-formulae of C.

More generally, we obtain a way to carry properties from the theory of C∗-algebras to thefirst order theory of complex numbers in boolean valued models of ZFC, and conversely.

1

Chapter 1

Functional Analysis

The aim of this chapter is to briefly present the tools we need from functional analysis. The finalintention is to define C∗-algebras and enunciate the Gelfand-Naimark Theorem. The referencetexts for this part, where to find details and proofs, are [2] and [11] for the general part offunctional analysis, and the first chapter of [4] for the part regarding C∗-algebras.

1.1 Banach Spaces and Weak Topologies

Since our aim is to present the theory of C∗-algebras, we will only work with vector spaces onthe complex field C.

Definition 1.1.1. Given a vector space V, a norm is a function:

‖.‖ : V → R

such that for x, y ∈ V, λ ∈ C:

• ‖x‖ ≥ 0,

• ‖x‖ = 0⇔ x = 0,

• ‖λx‖ = |λ|‖x‖,

• ‖x+ y‖ ≤ ‖x‖+ ‖y‖.

A normed vector space is a pair (V, ‖.‖V) (the norm could be omitted if no confusion canarise).

A distance can be defined on such spaces through the norm, as follows:

d(x, y) = ‖y − x‖

We can consider therefore the topology induced by this distance.Continuous linear functions between normed vector spaces have a strong characterization,

given by the following proposition:

Proposition 1.1.2. Assume T is a linear function between two normed spaces (V, ‖.‖V) and(U , ‖.‖U ). Then the following are equivalent:

2 Chapter 1. Functional Analysis

1. T is continuous;

2. T is uniformly continuous;

3. T is continuous at 0;

4. T is bounded, i. e. there is C > 0 such that for all x ∈ V ‖T (x)‖U ≤ C‖x‖V holds.

Proof. A proof can be found in [11, Lemma 4.1].

Definition 1.1.3. Given a normed vector space V, its dual space is the set:

V∗ = f : V → C | f is linear and continuous

Remark 1.1.4. The previous proposition tells us that, if f ∈ V∗, the value sup‖x‖≤1|f(x)| isbounded, hence we can define the following function:

‖.‖ : V∗ → Rf 7→ sup

‖x‖≤1

|f(x)|

and this happens to be a norm (see [11, Lemma 4.15]). This means that the dual space of anormed vector space is a normed vector space itself, and it is always complete in the topologyinduced by the norm ([11, Theorem 4.27]). This is an example of a Banach space.

Definition 1.1.5. A Banach space is a normed vector space (E , ‖.‖) which is complete for themetric induced by the norm.

Besides the topology induced by the norm, in functional analysis there are two other topologieswhich are often defined on Banach spaces (actually the second can be defined only on the dualof a Banach space).

Definition 1.1.6. Given a set X, a family of functions F = fii∈I , and a family of topologicalspaces Yii∈I such that fi : X → Yi, then the coarsest topology for X associated to F is thesmallest topology on X which renders each element in F continuous.

The proof of the existence of such a topology can be found in the first section of Chapter 3in [2], where it is also explained how to find a basis for this topology:

Proposition 1.1.7. A basis for the coarsest topology of X associated to F is given by the familyof finite intersections of sets in Uλλ∈Λ, where Uλ = f−1

i (V ) for some fi ∈ F and V open setin Yi

Definition 1.1.8. Given a Banach space E , the weak topology (usually denoted with σ(E , E∗))on E is the coarsest topology on E associated to E∗.

Remark 1.1.9. Since the functions in E∗ are continuous in the topology induced by the norm,the norm topology on E contains the weak topology σ(E , E∗).

Proposition 1.1.10. The weak topology on E is generated by the following basis of neighborhoods,as f1, . . . , fk ∈ E∗, ε ∈ R+ and x0 ∈ E vary:

Vx0(f1, . . . , fk; ε) = x ∈ E : |fi(x− x0)| < ε ∀i = 1, . . . , k .

1.1. Banach Spaces and Weak Topologies 3

A proof of this can be found section 3.2 of [2] (Proposition 3.4), with a deeper explanationof this topology, and all the details which have been omitted here.

We are more interested on a third topology which can be defined on the dual of a Banachspace.

Definition 1.1.11. The bidual space E∗∗ of a Banach space is the dual space of E∗.

The following map is an embedding of E into E∗∗:

J : E → E∗∗

x 7→ x

where, given f ∈ E∗, x(f) = f(x). The linearity of x follows from the linearity of f , and thecontinuity holds because of the following:

|x(f)| = |f(x)| ≤ ‖f‖‖x‖.

Definition 1.1.12. Given a Banach space E , the weak∗ topology on E∗ (denoted by σ(E∗, E))is the coarsest topology of E∗ associated to J(E).

Remark 1.1.13. If E is such that J is surjective, then the weak topology and the weak∗ topologyon E∗ overlap.

For our purposes, only few properties of this topology will be enunciated. All the facts wehave already stated and the properties we will present from now on about the weak∗ topologycan be found in Chapter 3, Section 4, in [2]. In particular, the reason why this topology isconsidered in analysis is briefly explained in Remark 8 of that Chapter.

Proposition 1.1.14. The weak∗ topology on E∗ is generated by the following base of neighbor-hoods, as x1, . . . , xk ∈ E, f0 ∈ E∗ and ε ∈ R+ vary:

Vf0(x1, . . . , xk; ε) = f ∈ E : |xi(f − f0)| < ε ∀i = 1, . . . , k .

Proposition 1.1.15. The space E∗ equipped with the weak∗ topology is an Hausdorff space.

An interesting aspect of this topology is that the convergence of a sequence to an element isthe pointwise convergence:

Proposition 1.1.16. A sequence (fn) in E∗ converges to f in the weak∗ topology if and only iffn(x) converges to f(x) for all x ∈ E.

In defining σ(E∗, E) many open set from the topology defined by the norm on E∗ were removed(the weak∗ topology is contained in the one induced by the norm). This operation was done inorder to have more compact sets and in fact we have the following result:

Theorem 1.1.17 (Banach-Alaoglu-Bourbaki). In the space E∗ with weak∗ topology, the closedball

B0(1) = f ∈ V∗ : ‖f‖ ≤ 1

is compact.

Proof. See [2, Theorem 3.16].

Remark 1.1.18. The theorem is in general false for the topology induced by the norm, since weare working with vector spaces which may have infinite dimension (see [11, Theorem 2.26]).

4 Chapter 1. Functional Analysis

1.2 C∗-algebras and Gelfand Transform

Definition 1.2.1. A C-algebra A with a norm (so that A is a normed vector space) is a Banachalgebra if it is a Banach space and for x, z ∈ A:

‖xz‖ ≤ ‖x‖‖z‖

A Banach algebra is unital if there is a neutral element for the product.

Definition 1.2.2. We call involution on a C-algebra A an operator ∗ : A → A such that forall x, y ∈ A and λ ∈ C:

• (x+ y)∗ = x∗ + y∗,

• (xy)∗ = y∗x∗,

• (λx)∗ = λx∗,

• x∗∗ = x.

A C-algebra A equipped with an involution is called a ∗-algebra.A C∗-algebra is a Banach ∗-algebra which satisfies the property

‖x∗x‖ = ‖x‖2

Remark 1.2.3. As we can see, all these definitions tend to generalize operations which are well-known in the field of complex numbers: C equipped with its usual product and with conjugationas involution is the most elementary example of C∗-algebra.

Definition 1.2.4. Let A and B be two C-algebras. A homomorphism from A to B is a mapϕ : A → B such that:

• ϕ is linear;

• ϕ bounded (hence continuous);

• ϕ(xy) = ϕ(x)ϕ(y) for each x, y ∈ A.

IfA and B are ∗-algebras, ϕ is said to be a ∗-homomorphism if it satisfies ϕ(x∗) = ϕ(x)∗ as well.An isomorphism of C-algebras is a bijective homomorphism whose inverse is an homomorphism.A ∗-isomorphism is a bijective ∗-homomorphism whose inverse is a ∗-homomorphism

There are many interesting examples of C∗-algebras which motivate the study of these objects,but we are essentially interested in the following one:

Example 1.2.5. Let X be an Hausdorff and compact space. Consider

C(X) = f : X → C | f is continuous

with pointwise sum and product and the uniform norm. If we also define f∗ = f , that isf∗(x) = f(x), it is easy to check that we obtain a C∗-algebra.

From now on, we will focus on commutative unital C∗-algebras. Gelfand theory starts froma specific object which can be defined for a Banach algebra A: its spectrum.

Definition 1.2.6. Given a Banach algebra A the set of non-zero homomorphisms from A to Cis the spectrum of A, and we denote it with σ(A).

1.2. C∗-algebras and Gelfand Transform 5

Proposition 1.2.7. Given a commutative unital Banach algebra A and h ∈ σ(A), the followingholds:

• h(e) = 1

• |h(x)| ≤ ‖x‖

Proof. See [4, Proposition 1.10].

Proposition 1.2.8. The spectrum σ(A) of a commutative unital Banach algebra is an Hausdorffcompact subspace of A∗ in the weak∗ topology.

Proof. Since |h(x)| ≤ ‖x‖, σ(A) is a subset of the ball of radius 1 centered in 0 in A∗. Thismeans that the spectrum can be defined equivalently as

σ(A) = h ∈ B1(0) : h(e) = 1 ∧ h(xy) = h(x)h(y) ∀x, y ∈ A .

Since the conditions h(e) = 1 and h(xy) = h(x)h(y) are preserved under pointwise limits, σ(A)is closed in B1(0), hence is an Hausdorff compact subspace of A∗ in the weak∗ topology.

Definition 1.2.9. The Gelfand transform of a Banach algebra A is the operator

ΓA : A → C(σ(A))

x 7→ x

where x(h) = h(x). We will often denote ΓA as Γ.

The Gelfand-Naimark Theorem shows a strong relation between a commutative unital C∗

and its spectrum.

Theorem 1.2.10 (Gelfand-Naimark). Assume A is a commutative and unital C∗-algebra, thenΓ is an isometric ∗-isomorphism from A to C(σ(A)).

Proof. See [4, Section 1.2, Theorem 1.20] for the details of the proof.

Remark 1.2.11. The Gelfand-Naimark Theorem tells us that, in the study of commutative unitalC∗-algebras, it is enough to focus on the function algebras of the form C(X) with X compactHausdorff space, since any commutative unital C∗-algebra has an isomorphic copy among thesetype of C∗-algebras.

7

Chapter 2

First order logic and BooleanValued Models

This chapter will be dedicated to mathematical logic, with a focus on boolean valued models andtheir semantics. Boolean valued models generalize the structure of first order models, combiningfirst order model theory with boolean algebras. We will first give a presentation of the basictools and theorems of model theory. Our exposition will be short, we refer to [9, Chapters 1,2]for further details. References for the part regarding the basic properties of boolean algebras are[5] and [7].

2.1 First order logic

Definition 2.1.1. A language is a set:

L = Ri : i ∈ I ∪ fj : j ∈ J ∪ ck : k ∈ K

where:

• every Ri is called relation symbol;

• every fj is called function symbol;

• every ck is called constant symbol;

with a functionΘL : Ri : i ∈ I ∪ fj : j ∈ J → N

which associates to each relation (function) symbol a number n which is called arity of therelation (function).

We will usually refer to a language omitting the function ΘL (if no confusion can arise).

Definition 2.1.2. Given a language

L = Ri : i ∈ I ∪ fj : j ∈ J ∪ ck : k ∈ K

a L-structure M is a tuple 〈M,RMi : i ∈ I, fMj : j ∈ J, cMk : k ∈ K〉 where:

• M is a non-empty set, called domain of the structure;

8 Chapter 2. First order logic and Boolean Valued Models

• RMi is a subset of Mn (where n is the arity of Ri) called interpretation of Ri in M;

• fMj is a function from Mn to M (where n is the arity of fj) called interpretation of fjin M;

• cMk is a element of M called interpretation of ck in M .

The relevant maps between L-structures are those which preserve the interpretations of sym-bols in L.

Definition 2.1.3. Given two L-structures M and N , an L-embedding is an injective mapϕ : M → N such that:

• given a relation symbol R ∈ L whose arity is n,

(a1, . . . , an) ∈ RM ⇔ (ϕ(a1), . . . , ϕ(an)) ∈ RN

• given a function symbol f ∈ L whose arity is n,

ϕ(fM(a1, . . . , an)) = fN (ϕ(a1), . . . , ϕ(an))

• given a constant symbol c ∈ L,ϕ(cM) = cN

An isomorphism is a bijective L-embedding.

Definition 2.1.4. If M ⊆ N andM and N are two L-structures such that the immersion of Min N is an L-embedding, we say thatM is a substructure of N , or that N is an extension ofM.

Remark 2.1.5. At this stage of model theory, the aim is to formalize the idea that a formula istrue in a certain structure. In order to do this, we need to formalize the notion of formula in alanguage L. Among the symbols we want to appear in a formula there are variables, hence, fromnow on, when we will consider a certain language L, we will also assume to have a countable setV = x1, x2, . . . of variables.

Definition 2.1.6. Given a language L, the set of L-terms is the smallest set TL such that:

• if c is a constant symbol in L, then c ∈ TL;

• if x ∈ V, then x ∈ TL;

• if t1, . . . , tn ∈ TL and f is a function symbol in L with arity n, then f(t1, . . . , tn) ∈ TL.

A closed term is a term in which no variables occur.

Definition 2.1.7. Given a language L, we say that ϕ is an atomic L-formula (or atomicformula) if it is one of the following:

• t1 = t2, where t1 and t2 are L-terms;

• R(t1, . . . , tn) where R is relation symbol in L of arity n and ti are terms.

The set of all L-formulae (or formulae) is the smallest set FL such that it contains the atomicL-formulae and such that:

2.1. First order logic 9

• if ϕ ∈ F then ¬ϕ ∈ F ;

• if ϕ,ψ ∈ F then ϕ ∧ ψ ∈ F ;

• if ϕ ∈ F then ∃xϕ ∈ F .

We will also use the following abbreviations:

• ϕ ∨ ψ ≡ ¬(¬ϕ ∧ ¬ψ);

• ϕ→ ψ ≡ ¬ϕ ∨ ψ;

• ϕ↔ ψ ≡ (ϕ→ ψ) ∧ (ψ → ϕ);

• ∀xϕ ≡ ¬∃x¬ϕ.

Remark 2.1.8. Given a formula ϕ, an occurrence of a variable x is bound if in such occurrencex is immediately after a quantification symbol ∃ or ∀. An occurrence of x is free if is not bound.The set of free variables of a formula ϕ is the set of those variables which have at least onefree occurrence in ϕ. If the set of free variables of ϕ is contained in x1, . . . , xn we will writeϕ(x1, . . . , xn) instead of ϕ. If a L-formula has no free variables, we will call it a L-statement(or a statement).

Definition 2.1.9. Let x1, . . . , xn be a set of variables. A valuation in an L-structure M ofthese variables is a function

ν : x1, . . . , xn →M

If ν(xi) = ai, we will also write ν = (x1/a1, . . . , xn/an).Given a formula ϕ(x1, . . . , xn), with ϕ(ν) or ϕ(a1, . . . , an) we will denote the formula ϕ in

which every occurrence of a variable xi is substituted with ν(xi) = ai. We obtain this way aformula with parameters in M.

Definition 2.1.10. Given an L-term t whose variables are included in the domain of a valuationν in M, we inductively define t(ν) ∈M as follows:

• if t is a constant c, then t(ν) = c;

• if t is a variable x, then t(ν) = ν(x);

• if t = f(t1, . . . , tn), where f is a function symbol in L of arity n and ti are L-terms, thent(ν) = f(t1(ν), . . . , tn(ν)).

Definition 2.1.11 (Tarski’s Semantic). Given an L-structure M, let ϕ be a formula in thelanguage L whose free variables are in x1, . . . , xn, and ν a valuation in M whose domaincontains x1, . . . , xn. We say that ϕ(ν) is true in M, or M |= ϕ(ν), in the following cases (byrecursion):

• if ϕ ≡ t1 = t2, then M |= ϕ(ν) iff t1(ν) = t2(ν);

• if ϕ ≡ R(t1, . . . , tn), then M |= ϕ(ν) iff (t1(ν), . . . , tn(ν)) ∈ RM;

• if ϕ ≡ ¬ψ, then M |= ϕ(ν) iff M 6|= ψ(ν)

• if ϕ ≡ ψ ∧ θ, then M |= ϕ(ν) iff M |= ψ(ν) and M |= θ(ν);

• if ϕ ≡ ∃yψ(y), then M |= ϕ(ν) iff there is b ∈M such that M |= ψ(y/b, ν).


Remark 2.1.12. If ϕ is a statement, the previous definition does not require any valuation inorder to decide if M |= ϕ.

Definition 2.1.13. Given a language L, a L-theory (or just theory) is a set T of L-statements.We say that a theory is consistent if there is a L-structure M such that for each ϕ ∈ T ,

M |= ϕ holds (we might also write M |= T ).We say that a L-statement ϕ is a logical consequence of a L-theory T if

M |= T ⇒M |= ϕ

Remark 2.1.14. We can give another notion of logical consequence, assuming some rules whichtell us how to obtain a formula from other formulae, as we do in a proof. A collection of suchrules is called a proof system, and we will refer to the one defined in [12] (Chapter 2, Section6).

Logical axioms:

– ϕ ∨ ¬ϕ;

– x = x;

– ϕ(a)→ ∃ϕ(x);

– (x = y)→ (f(x) = f(y))

– (x = y)→ (ϕ(x)→ ϕ(y)).

Rules of inference:

– ϕ ` ϕ ∨ ψ;

– ϕ ∨ ϕ ` ϕ;

– (ϕ ∨ (ψ ∨ χ)) ` ((ϕ ∨ ψ) ∨ χ);

– (ϕ ∨ ψ) ∧ (¬ϕ ∨ χ) ` ψ ∨ χ;

– if x is not free in ψ, ϕ(a)→ ψ ` ∃x(ϕ(x)→ ψ);

We will say that a statement ψ is syntactically provable from a theory T , and denote itwith T ` ψ, if there is a finite sequence of formulae ϕ1, . . . , ϕk such that ϕk = ψ and for each1 ≤ i ≤ k one of the following holds:

• ϕi ∈ T ;

• ϕi is a logical axiom;

• ϕi can be obtained from ϕ1, . . . , ϕi−1 through a rule of inference.

Theorem 2.1.15 (Soundness Theorem). Given an L-theory T and an L-statement ϕ, if T ` ϕthen T |= ϕ.

The vice versa holds as well.

Theorem 2.1.16 (Completeness Theorem). Given an L-theory T and an L-statement ϕ, ifT |= ϕ then T ` ϕ.

Proof. A proof of both theorems can be found in [12] (Chapter 2 and 4 respectively).

We conclude this part presenting the most important theorems of elementary model theory.The omitted proofs can be found in [9, Chapter 2, Sections 1 and 3].

2.2. Boolean Algebras 11

Theorem 2.1.17 (Compactness Theorem). A theory T is consistent if and only if every finitesubset of T is consistent.

Definition 2.1.18. Given two L-structures M and N , we say that they are elementarilyequivalent, and denote it with M≡ N , if for every L-statement ϕ

M |= ϕ⇔ N |= ϕ

We say thatM is an elementary substructure of N , denoting it withM≺ N , if M ⊆ Nand given a L-formula ϕ(x1, . . . , xn) with (a1, . . . , an) ∈Mn, then:

M |= ϕ(a1, . . . , an)⇔ N |= ϕ(a1, . . . , an)

Let T be a set of L-formulae. We will write

M≺T N

if the property above holds only for ϕ ∈ T .

Theorem 2.1.19 (Upward Lowenheim−Skolem Theorem). Let M be an infinite L-structureand κ a cardinal such that κ ≥ |L|+ |M |. Then there is an L-structure N such that |N | = κ andM≺ N .

Theorem 2.1.20 (Downward Lowenheim−Skolem Theorem). Let M be an L-structure andA ⊆M . There is N such that A ⊆ N , |N | ≤ |A|+ |L|+ ℵ0 and N ≺M.

2.2 Boolean Algebras

As mentioned earlier, good references for the material of this part are [5] and [7].

Definition 2.2.1. A partial order (B, <) is called a boolean algebra if:

• every two elements a, b ∈ B admit a unique least upper bound a ∨ b;

• every two elements a, b ∈ B admit a unique greatest lower bound a ∧ b;

• there are two elements 0B, 1B ∈ B such that for every a ∈ B it holds 0B ≤ a ≤ 1B;

• is distributive, which means

a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)

a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)for every a, b, c ∈ B;

• is complemented, that is for every a ∈ B there is a unique ¬a such that a∧¬a = 0B anda ∨ ¬a = 1B.

A boolean algebra is complete if any ai : i ∈ I subset of B admits a supremum∨i∈I ai and

an infimum∧i∈I ai.

Remark 2.2.2. Given a, b ∈ B a boolean algebra, we have that:

a ≤ b⇔ a ∧ b = a

a ≤ b⇔ a ∨ b = b


Example 2.2.3. Given a topological space X, CL(X) is the set of all clopen subsets of X.CL(X) is a boolean algebra with the following operations:

A ≤ B ⇔ A ⊆ B

A ∧B = A ∩B

A ∨B = A ∪B

¬A = X \A

Another example is given by the set RO(X), the collection of all regular open subsets of a

topological space X. An open subset A of X is regular if A = A. With the following operations,RO(X) is a complete boolean algebra:

A ≤ B ⇔ A ⊆ B

A ∧B = A ∩B

A ∨B = ˚A ∪B

¬A = ˚(X \A)∨i∈I

Ai =⋃Ai

∧i∈I

Ai =⋂Ai

A proof of this can be found in [5, Chapter 10].

Definition 2.2.4. A morphism between two boolean algebras B and C is a map ϕ : B → Csuch that, given a, b ∈ B:

• ϕ(0B) = 0C and ϕ(1B) = 1C;

• if a ≤ b then ϕ(a) ≤ ϕ(b);

• ϕ(a ∧ b) = ϕ(a) ∧ ϕ(b);

• ϕ(a ∨ b) = ϕ(a) ∨ ϕ(b);

• ϕ(¬a) = ¬ϕ(a).

An isomorphism of boolean algebras is a bijective morphism. A morphism is complete ifpreserves all suprema and infima.

Lemma 2.2.5. Every isomorphism is complete.

Proof. Let i : B→ C an isomorphism and consider aj : j ∈ J ⊆ B. We want to show

i

∨j∈J

aj

=∨j∈J

i(aj)


On the one hand∨j∈J aj ≥ aj , hence i(

∨j∈J aj) ≥

∨j∈J i(aj). On the other hand, if b ≥ i(aj)

for every j ∈ J , let a ∈ B such that b = i(a). It follows a ≥ aj for each j ∈ J , hence a ≥∨j∈J aj ,

and in conclusion

b = i(a) ≥ i

∨j∈J

aj

The proof for infima is similar.

We now turn to the topological counterparts of a boolean algebra, its Stone space.

Definition 2.2.6. Given B a boolean algebra, a filter is a subset F of B such that:

• 1B ∈ F and 0B /∈ F ;

• if a ∈ F and b ∈ F then a ∧ b ∈ F ;

• if a ∈ F and a ≤ b then b ∈ F .

A ultrafilter G is a filter such that one of the following equivalent properties holds:

• if a /∈ G then ¬a ∈ G;

• if a1 ∨ · · · ∨ ak ∈ G then there is a ai such that ai ∈ G.

The Stone space of B is the set

St(B) = G ⊂ B : G is a ultrafilter

Remark 2.2.7. It follows from the definitions that for a given ultrafilter G on the boolean algebraB, a ∈ G⇔ ¬a /∈ G.

Remark 2.2.8. Let i : B→ C be an injective morphism of boolean algebras, and let F be a filterin B. We can define the set:

i∗(F ) = a ∈ C : ∃b ∈ F such that a ≥ i(b)

Since the only b ∈ B such that its image is 0C is 0B, i∗(F ) can be easily verified to be a filter.

Theorem 2.2.9 (Maximal Ideal Theorem). If F is a filter in a boolean algebra B, then F canbe extended to a ultrafilter in B.

Proof. A proof of this theorem can be found in [7, Chapter 2, Proposition 2.16].

Definition 2.2.10. Let B be a boolean algebra and F ⊆ B a filter. The following is an equiva-lence relation on B:

a ∼F b⇔ ∃f ∈ F : a ∧ f = b ∧ f

The set of all equivalence classes B/F is called quotient algebra, and it is a boolean algebrawith the following order relation and operations:

• [a]F ≤ [b]F ⇔ a < b;

• [a ∧ b]F = [a]F ∧ [b]F ;

• [a ∨ b]F = [a]F ∨ [b]F ;

• ¬[a]F = [¬a]F .


Moreover 1B/F = [1B]F and 0B/F = [0B]F (a proof of all these facts can be found in [7, Lemmaand Definition 5.22]).

Proposition 2.2.11. Let B and C be two boolean algebras and i : B→ C an injective morphism.Assume F is a filter in B, then the map

iF : B/F → C/i∗(F )

[a]F 7→ [i(a)]i∗(F )

is a well-defined injective morphism of boolean algebras. Moreover, if i is an isomorphism, iF isan isomorphism as well.

Proof. First, we show that iF is well-defined. Let a, a′ ∈ B such that a ∼F a′, hence there isf ∈ F such that a ∧ f = a′ ∧ f . This implies

i(a) ∧ i(f) = i(a ∧ f) = i(a′ ∧ f) = i(a′) ∧ i(f)

so that i(a) ∼i∗(F ) i(a′) and iF is a well-defined map.

The map iF is a morphism, in fact:

• we have that

iF (0B/F ) = iF ([0B]F ) = [i(0B)]i∗(F ) = [0C]i∗(F ) = 0C/i∗(F )

andiF (1B/F ) = iF ([1B]F ) = [i(1B)]i∗(F ) = [1C]i∗(F ) = 1C/i∗(F )

• if [a]F ≤ [b]F ∈ B/F :

iF ([a]F ) = [i(a)]i∗(F ) ≤ [i(b)]i∗(F ) = iF ([b]F )

since a ≤ b and i is a morphism itself;

• if [a]F , [b]F ∈ B/F :

iF ([a]F ∧ [b]F ) = iF ([a ∧ b]F ) = [i(a ∧ b)]i∗(F ) = [i(a) ∧ i(b)]i∗(F ) =

[i(a)]i∗(F ) ∧ [i(b)]i∗(F ) = iF ([a]F ) ∧ iF ([b]F )

• if [a]F , [b]F ∈ B/F :

iF ([a]F ∨ [b]F ) = iF ([a ∨ b]F ) = [i(a ∨ b)]i∗(F ) = [i(a) ∨ i(b)]i∗(F ) =

[i(a)]i∗(F ) ∨ [i(b)]i∗(F ) = iF ([a]F ) ∨ iF ([b]F )

• if [a]F ∈ B/F :

iF (¬[a]F ) = iF ([¬a]F ) = [i(¬a)]i∗(F ) = [¬i(a)]i∗(F ) = ¬[i(a)]i∗(F ) = ¬iF ([a]F )

The morphism iF is injective since assuming i(a) ∼i∗(F ) i(b), it can be found f ∈ i∗(F ) suchthat

i(a) ∧ f = i(b) ∧ fThis means that we can find c ∈ F such that f ≥ i(c), so that

i(a) ∧ i(c) = i(a) ∧ (i(c) ∧ f) = i(b) ∧ (i(c) ∧ f) = i(b) ∧ i(c)

Since i is injective this implies a ∧ c = b ∧ c, thus a ∼F b.Assume i is surjective. Then iF is surjective as well, since given b ∈ N it can be found a

a ∈M such that i(a) = b, hence [b]i∗(F ) = iF ([a]F ).


Remark 2.2.12. If G is a ultrafilter in B, then B/G = [0B ]G, [1B ]G. In fact, there are only twoclasses: the one containing all the elements in G, and the one containing their complements (i.e. all the elements not in G).

If a ∈ G, then a ∧ a = 1B ∧ a, which leads to a ∼G 1B (this is true for any filter F ). On theother side, if a /∈ G, then ¬a ∈ G, and since a ∧ ¬a = 0B ∧ ¬a, a ∼G 0B holds.

On St(B) we define the topology generated by the family of sets Oaa∈B, where:

Oa = G ∈ St(B) : a ∈ G

Proposition 2.2.13. Given a boolean algebra B, St(B) is an Hausdorff compact space, and thefamily Oaa∈B is a base of clopen sets (i.e. St(B) is 0-dimensional).

Proof. A proof of this proposition can be found in [7, Theorem 7.8].

Theorem 2.2.14 (Stone’s Representation Theorem). Every boolean algebra B is isomorphic tothe boolean algebra CL(St(B)).

Proof. The map

Φ : B→ CL(St(B))

a 7→ Oa

is an isomorphism of boolean algebras. The details of the proof can be found in [7, Theorems2.1 and 7.8].

Remark 2.2.15. With the following proposition we will characterize the completeness of a booleanalgebra with the topology of its Stone space. In particular, a topological space X is extremelydisconnected if RO(X) = CL(X).

Proposition 2.2.16. A boolean algebra B is complete ⇔ St(B) is extremely disconnected.

Proof. A proof of this proposition can be found in [7, Proposition 7.21].

The following proposition gives a characterization of 0-dimensional Hausdorff compact spacesin terms of the Stone space of a specific boolean algebra. We will give an explicit proof of itsince we will need the map defined in it later on.

Proposition 2.2.17. Let X be a 0-dimensional compact Hausdorff topological space, then X ishomeomorphic to St(CL(X)).

Proof. We shall define the map:

ϕ : X → St(CL(X))

x 7→ Gx

where Gx = A ∈ CL(X) : x ∈ A can be easily checked to be a ultrafilter.

Injective: If x 6= y, let A be a clopen set such that x ∈ A, y ∈ Ac (we can find such set since Xis 0-dimensional and Hausdorff). A ∈ Gx and Ac ∈ Gy, thus these ultrafilters have to bedifferent.


Surjective: Let G ∈ St(CL(X)); it is enough to show that C =⋂G is a singleton. In fact,

C = z implies that if A ∈ G then z ∈ A. On the other hand, if B is clopen anda /∈ B, it cannot be B ∈ G (otherwise a /∈ C), hence z ∈ Bc ∈ G. This means thatA ∈ G ⇔ z ∈ A ⇔ A ∈ Gz, so that G = Gz. We have that C is non-empty, because Ghas the finite intersection property and X is compact. Let x 6= y be in C, we can thereforefind a clopen set A such that x ∈ A and y /∈ A. It follows that

A ∈ G⇔ Ac /∈ G

which means

x ∈ C ⇔ y /∈ C

and this is absurd.

Continuous and open: Let A be a clopen set in X. x ∈ A if and only if A ∈ Gx, which meansϕ(A) = OA. It follows that the map ϕ is open and continuous.

Proposition 2.2.18. Let B and C be two boolean algebras, and f be a continuous map:

f : St(C)→ St(B)

We can define the following morphism of boolean algebras:

f∗ : B→ C

a 7→ f−1[a]

where we are identifying the elements of a boolean algebra with the clopen sets of their associatedStone spaces as in Theorem 2.2.14.

Given another boolean algebra D, and a continuous map

g : St(B)→ St(D)

the following properties hold:

1. (g f)∗ = f∗ g∗;

2. if B = C and f = id, then id∗ = idB;

3. if f is an homeomorphism, then (f∗) is an isomorphism of boolean algebras and (f∗)−1 =(f−1)∗.

Proof. A proof of this can be found in [7, Theorem 8.2].

Remark 2.2.19. The last proposition tells us that if two boolean algebras have homeomorphicStone spaces, then they are isomorphic. Since the converse is trivial, we have the following:

Corollary 2.2.20. The boolean algebras B and C are isomorphic if and only if St(B) and St(C)are homeomorphic.

2.3. Boolean Valued Models 17

2.3 Boolean Valued Models

In a first order model a formula can be interpreted as true or false. Given a complete booleanalgebra B, B-boolean valued models generalize Tarski semantics associating to each formula avalue in B, so that there are no more only true and false propositions (those associated to 1Band 0B respectively), but also other “intermediate values” (even not comparable values, since Bis not necessarily a linear order) of truth. This section presents and extends some of the contentsof the notes [14] regarding boolean valued models.

Definition 2.3.1. Given a complete boolean algebra B, and a first order language

L = Ri : i ∈ I ∪ fj : j ∈ J ∪ ck : k ∈ K ,

a B-boolean valued model (or B-valued model) M in the language L is a tuple

〈M,=M, RMi : i ∈ I, fMj : j ∈ J, cMk : k ∈ K〉

where:

1. M is a non-empty set, called domain of the B-boolean valued model, whose elements arecalled B-names;

2. =M is the boolean value of the equality:

=M: M2 → B

(τ, σ) 7→ Jτ = σKMB

3. RMi is the boolean interpretation of the n-ary relation symbol Ri:

RMi : Mn → B

(τ1, . . . , τn) 7→ JRi(τ1, . . . , τn)KMB

4. fMj is the boolean interpretation of the n-ary function symbol fj :

fMj : Mn+1 → B

(τ1, . . . , τn, σ) 7→ Jfj(τ1, . . . , τn) = σKMB

5. cMk is the boolean interpretation of the constant symbol ck, and it is an element in M .

We require that the following conditions hold:

for τ, σ, χ ∈M ,

i) Jτ = τKMB = 1B;

ii) Jτ = σKMB = Jσ = τKMB ;

iii) Jτ = σKMB ∧ Jσ = χKMB ≤ Jτ = χKMB ;

for Ri ∈ L with arity n, and (τ1, . . . , τn), (σ1, . . . , σn) ∈Mn,

iv) (∧h∈1,...,n Jτh = σhK

MB ) ∧ JRi(τ1, . . . , τn)KMB ≤ JRi(σ1, . . . , σn)KMB ;


for fj ∈ L with arity n, and (τ1, . . . , τn), (σ1, . . . , σn) ∈Mn and µ, ν ∈M ,

v) (∧h∈1,...,n Jτh = σhK

MB ) ∧ Jfj(τ1, . . . , τn) = µKMB ≤ Jfj(σ1, . . . , σn) = µKMB ;

vi)∨µ∈M Jfj(τ1, . . . , τn) = µKMB = 1B;

vii) Jfj(τ1, . . . , τn) = µKMB ∧ Jfj(τ1, . . . , τn) = νKMB ≤ Jµ = νKMB .

If no confusion can arise, we will omit the pedix B and we will confuse a function or predicatesymbol with its interpretation.

Remark 2.3.2. Every first order model naturally defines a B-valued model with B = 0, 1.

Remark 2.3.3. Given τ = (τ1, . . . , τn) and σ = (σ1, . . . , σn) in Mn, we will often use the followingabbreviation

Jτ = σKM ≡∧

h∈1,...,n

Jτh = σhKM

Definition 2.3.4. Let M be a B-valued model and N a C-valued model in the same languageL. Let

i : B→ C

be a morphism of boolean algebras and Φ ⊆M ×N a relation. The couple 〈i,Φ〉 is a morphismof boolean valued models if:

1. domΦ = M ;

2. given (τ1, σ1), (τ2, σ2) ∈ Φ:

i(Jτ1 = τ2KMB ) ≤ Jσ1 = σ2K

NC ,

3. given R an n-ary relation symbol and (τ1, σ1), . . . , (τn, σn) ∈ Φ:

i(JR(τ1, . . . , τn)KMB ) ≤ JR(σ1, . . . , σn)KNC ,

4. given f an n-ary function symbol and (τ1, σ1), . . . , (τn, σn), (µ, ν) ∈ Φ:

i(Jf(τ1, . . . , τn) = µKMB ) ≤ Jf(σ1, . . . , σn) = νKNC ,

5. given a constant symbol c and (τ, σ) ∈ Φ:

i(Jτ = cKMB ) ≤ Jσ = cKNC .

An injective morphism is a morphism such that in 2 equality holds.

An embedding of boolean valued models is an injective morphism such that in 3-5 equalityholds.

An embedding 〈i,Φ〉 from M to N is called isomorphism of boolean valued models if i isan isomorphism of boolean algebras and for every b ∈ N there is a a ∈M such that (a, b) ∈ Φ.

Remark 2.3.5. When B = C we will consider i = idB unless otherwise stated.


Definition 2.3.6. Suppose M is a B-valued model and N a C-valued model (both in the samelanguage L) such that B is a complete subalgebra of C, M ⊆ N , and

JR(τ1, . . . , τn)KMB = JR(τ1, . . . , τn)KNC

Jf(τ1, . . . , τn) = σKMB = Jf(τ1, . . . , τn) = σKNC

Jτ = cKMB = Jτ = cKNCfor all relation symbols R, all function symbols f , and all constant symbols c in L. Let J be theimmersion of M into N . Then 〈idB, J〉 is an embedding of boolean valued models and N is saidto be a boolean extension of M.

Remark 2.3.7. The definition of valuation of variables in a first order model can be easilygeneralized to B-valued models, as the notion of formula with parameters. We will therefore usethe notations defined in Definition 2.1.9 for B-valued models as well.

Definition 2.3.8. Given a B-valued modelM in a language L, let ϕ be a L-formula whose freevariables are in x1, . . . , xn, and let ν be a valuation inM whose domain contains x1, . . . , xn.We define now Jϕ(ν)KMB , the boolean value of ϕ(ν).

First, let t be an L-term and τ ∈M ; we define recursively J(t = τ)(ν)KMB ∈ B as follows:

• if t is a constant c, then

J(c = τ)(ν)KMB =qcM = τ

yMB

• if t is a variable x, thenJ(x = τ)(ν)KMB = Jν(x) = τKMB

• if t = f(t1, . . . , tn) where ti are terms and f is an n-ary function symbol, then

J(f(t1, . . . , tn) = τ)(ν)KMB =∨

σ1,...,σn∈M

∧1≤i≤n

J(ti = σi)(ν)KMB

∧ Jf(σ1, . . . , σn) = τKMB

Given a formula ϕ, we define recursively Jϕ(ν)KMB as follows:

• if ϕ ≡ t1 = t2, then

J(t1 = t2)(ν)KMB =∨τ∈M

J(t1 = τ)(ν)KMB ∧ J(t2 = τ)(ν)KMB

• if ϕ ≡ R(t1, . . . , tn), then

J(R(t1, . . . , tn))(ν)KMB =∨

τ1,...,τn∈M

∧1≤i≤n

J(ti = τi)(ν)KMB

∧ JR(τ1, . . . , τn)KMB

• if ϕ ≡ ¬ψ, thenJϕ(ν)KMB = ¬ Jψ(ν)KMB

• if ϕ ≡ ψ ∧ θ, thenJϕ(ν)KMB = Jψ(ν)KMB ∧ Jθ(ν)KMB


• if ϕ ≡ ∃yψ(y), then

Jϕ(ν)KMB =∨τ∈M

Jψ(y/τ, ν)KMB

If no confusion can arise, we omit the index M and the pedix B, and we simply denote theboolean value of a formula ϕ with parameters in M by JϕK.

Remark 2.3.9. In the previous definition we considered the sup of several subsets of B, and thismotivates our request to work with a complete boolean algebra. If we assume that the languagedoes not contain function symbols, the definition of the boolean value of atomic formulae issimpler:

J(t1 = t2)(ν)K = Jt1(ν) = t2(ν)K

J(R(t1, . . . , tn))(ν)K = JR(t1(ν), . . . , tn(ν))K

where for a term t which is a constant or a variable, t(ν) can be defined verbatim for B-valuedmodels as for first order models. In this case we do not require B to be complete for the definitionof the boolean value of atomic formulae, but a certain degree of completeness of B is necessaryto give a semantic interpretation of the existential quantifier.

Proposition 2.3.10. Let M be a B-valued model and N a C-valued model in the same languageL. Assume 〈i,Φ〉 is an isomorphism of boolean valued models.

Then for any L-formula ϕ(x1, . . . , xn), and for every (τ1, σ1), . . . , (τn, σn) ∈ Φ we have that:

i(Jϕ(τ1, . . . , τn)KMB ) = Jϕ(σ1, . . . , σn)KNC

Proof. The proof proceeds by induction on the complexity of the formula. We will write ν =(x1/τ1, . . . , xn/τn) and ν′ = (x1/σ1, . . . , xn/σn) with (τi, σi) ∈ Φ.

Atomic formulae: First, consider (η, ζ) ∈ Φ and t an L-term. We will show

i(J(t = η)(ν)KMB ) = J(t = ζ)(ν′)KNC

If t is a constant or a variable, this follows because i is an injective morphism. If t =f(t1, . . . , tn) we use the fact that i is complete (see Lemma 2.2.5) and that 〈i,Φ〉 is anembedding and proceed by induction on the terms ti:

i(J(f(t1, . . . , tn) = η)(ν)KMB ) =

∨χ∈Mn

∧1≤i≤n

i(J(ti = χi)(ν)KMB )

∧ i(Jf(χ1, . . . , χn) = ηKMB )

Since Im(Φ) = N we have:

∨χ∈Mn

∧1≤i≤n


∧ i(Jf(χ1, . . . , χn) = ηKMB ) =

∨ω∈Nn

∧1≤i≤n

J(ti = ωi)(ν′)KNC

∧ Jf(ω1, . . . , ωn) = ζKNC =

J(f(t1, . . . , tn) = ζ)(ν′)KNC


Let now ϕ ≡ t1 = t2. By induction we have:

i(J(t1 = t2)(ν)KMB ) =∨χ∈M

i(J(t1 = χ)(ν)KMB ) ∧ i(J(t2 = χ)(ν)KMB )

From Im(Φ) = N follows that:∨χ∈M

i(J(t1 = χ)(ν)KMB ) ∧ i(J(t2 = χ)(ν)KMB ) =

∨ω∈N

J(t1 = ω)(ν′)KNC ∧ J(t2 = ω)(ν′)KNC = J(t1 = t2)(ν′)KNC

If ϕ ≡ R(t1, . . . , tn) we have:

i(JR(t1, . . . , tn)(ν)KMB ) =

∨χ∈Mn

∧1≤i≤n


∧ i(JR(χ1, . . . , χn)(ν)KMB ) =

∨ω∈Nn

∧1≤i≤n

J(ti = ωi)(ν′)KNC

∧ JR(ω1, . . . , ωn)(ν′)KNC = JR(t1, . . . , tn)(ν′)KNC

Negation: Consider ϕ ≡ ¬ψ. We have that:

i(Jϕ(ν)KMB ) = ¬i(Jψ(ν)KMB ) = ¬ Jψ(ν′)KNC = Jϕ(ν′)KNC

Conjunction: If ϕ ≡ ψ ∧ θ then:

i(Jϕ(ν)KMB ) = i(Jψ(ν)KMB ) ∧ i(Jθ(ν)KMB ) = Jψ(ν′)KNC ∧ Jθ(ν′)KNC = Jϕ(ν′)KNC

Existential: Let ϕ ≡ ∃yψ(y). We have that:

i(Jϕ(ν)KMB ) =∨χ∈M

i(Jψ(y/χ, ν)KMB ) =∨ω∈N

Jψ(y/ω, ν′)KNC = Jϕ(ν′)KNC

where second equality holds since 〈i,Φ〉 is an isomorphism.

Now we want to generalize the Soundness Theorem 2.1.15 to boolean valued models. In orderto do this we first need a:

Lemma 2.3.11. Given a B-valued model M in the language L, assume ϕ(x1, . . . , xn) is anL-formula and τ = (τ1, . . . , τn), σ = (σ1, . . . , σn) ∈Mn. Then:

Jτ = σK ∧ Jϕ(τ)K ≤ Jϕ(σ)K

Proof. The proof proceeds by induction on the complexity of ϕ. Given ϕ(x1, . . . , xn), we canconsider ν = (x1/τ1, . . . , xn/τn) and ν′ = (x1/σ1, . . . , xn/σn), so that the thesis becomes:

Jτ = σK ∧ Jϕ(ν)K ≤ Jϕ(ν′)K


Atomic formulae: If ϕ is an atomic formula the thesis follows from the definitions and from [5,Lemma 3, Chapter 8]. Here is the proof: first, consider a term t and µ ∈M . We will show,recursively on the complexity of the term, that

Jτ = σK ∧ J(t = µ)(ν)K ≤ J(t = µ)(ν′)K

If t is a constant or a variable, this follows from the definition of B-valued model. Ift = f(t1, . . . , tn) and the thesis holds for t1, . . . , tn, then we have:

Jτ = σK ∧ J(f(t1, . . . , tn) = µ)(ν)K =∨χ∈Mn

(q(t = χ)(ν)

y∧ Jτ = σK

)∧ Jf(χ1, . . . , χn) = µK ≤

∨χ∈Mn

q(t = χ)(ν′)

y∧ Jf(χ1, . . . , χn) = µK = J(f(t1, . . . , tn) = µ)(ν′)K

Consider now ϕ ≡ t1 = t2. We have then:

Jτ = σK ∧ J(t1 = t2)(ν)K =∨χ∈M

(J(t1 = χ)(ν)K ∧ Jτ = σK) ∧ (J(t2 = χ)(ν)K ∧ Jτ = σK) ≤

∨χ∈M

J(t1 = χ)(ν′)K ∧ J(t2 = χ)(ν′)K = J(t1 = t2)(ν′)K

If ϕ ≡ R(t1, . . . , tn), then:

Jτ = σK ∧ JR(t1, . . . , tn)(ν)K =∨

χ∈Mn

q(t = χ)(ν)

y∧ Jτ = σK ∧ JR(χ1, . . . , χn)K ≤

∨χ∈Mn

q(t = χ)(ν′)

y∧ JR(χ1, . . . , χn)K = JR(t1, . . . , tn)(ν′)K

Negation: If ϕ ≡ ¬ψ, by induction we have

Jτ = σK ∧ Jψ(ν)K ≤ Jψ(ν′)K

Jσ = τK ∧ Jψ(ν′)K ≤ Jψ(ν)K ,

which meansJτ = σK ∧ Jψ(ν)K ∧ Jψ(ν′)K = Jτ = σK ∧ Jψ(ν)K

andJτ = σK ∧ Jψ(ν)K ∧ Jψ(ν′)K = Jτ = σK ∧ Jψ(ν′)K .

HenceJτ = σK ∧ Jψ(ν′)K = Jτ = σK ∧ Jψ(ν)K

It follows that:

Jτ = σK ∧ ¬ Jψ(ν)K = Jτ = σK ∧ ¬ Jψ(ν)K ∧ (Jψ(ν′)K ∨ ¬ Jψ(ν′)K) =

(Jτ = σK ∧ Jψ(ν′)K ∧ ¬ Jψ(ν)K) ∨ (Jτ = σK ∧ ¬ Jψ(ν′)K ∧ ¬ Jψ(ν)K) =

(Jτ = σK ∧ Jψ(ν)K ∧ ¬ Jψ(ν)K) ∨ (Jτ = σK ∧ ¬ Jψ(ν′)K ∧ ¬ Jψ(ν)K) =

Jτ = σK ∧ ¬ Jψ(ν′)K ∧ ¬ Jψ(ν)K ,

which meansJτ = σK ∧ ¬ Jψ(ν)K ≤ ¬ Jψ(ν′)K .


Conjunction: If ϕ ≡ ψ ∧ θ we have:

Jτ = σK ∧ Jϕ(ν)K = (Jτ = σK ∧ Jψ(ν)K) ∧ (Jτ = σK ∧ Jθ(ν)K) ≤

Jψ(ν′)K ∧ Jθ(ν′)K = Jϕ(ν′)K

Existential: If ϕ ≡ ∃yψ(y) we have that:

Jτ = σK ∧ Jϕ(ν)K =∨χ∈M

(Jψ(y/χ, ν)K ∧ Jτ = σK) ≤∨χ∈M

Jψ(y/χ, ν′)K = Jϕ(ν′)K

Definition 2.3.12. LetM be a B-valued model in a language L. A formula ϕ with parametersinM is said to be satisfied or valid inM if JϕK = 1B. A theory T is valid inM if every ϕ ∈ Tis valid in M.

Remark 2.3.13. Given a boolean algebra B and two elements a, b in it, we can define a → b ≡¬a ∨ b, and we observe that

a→ b = 1B ⇔ a ≤ b.

On the one hand assume ¬a ∨ b = 1B. Then:

a = 1B ∧ a = a ∧ (¬a ∨ b) = (a ∧ ¬a) ∨ (a ∧ b) = 0 ∨ (a ∧ b) = a ∧ b

which is equivalent to a ≤ b. On the other hand a ≤ b implies 1B = ¬a ∨ a ≤ ¬a ∨ b.

Theorem 2.3.14 (Soundness Theorem). Let L be a language, if ϕ is a L-formula which issyntactically provable by a L-theory T , and T is valid in a B-valued modelM, then Jϕ(ν)KM = 1Bfor all valuations ν in M.

Proof. We refer to the system proof defined in [12, Chapter 2, Section 6]. We need to show that,for all valuations, all the logical axioms:

1. ϕ ∨ ¬ϕ;

2. x = x;

3. ϕ(a)→ ∃ϕ(x);

4. (x = y)→ (f(x) = f(y))

5. (x = y)→ (ϕ(x)→ ϕ(y));

have boolean value equal to 1B, and that for all logical rules:

6. ϕ ` ϕ ∨ ψ;

7. ϕ ∨ ϕ ` ϕ;

8. (ϕ ∨ (ψ ∨ χ)) ` ((ϕ ∨ ψ) ∨ χ);

9. (ϕ ∨ ψ) ∧ (¬ϕ ∨ χ) ` ψ ∨ χ;

10. if x is not free in ψ, ϕ(a)→ ψ ` ∃x(ϕ(x)→ ψ);

it holds that if ϕ ` ψ, then JϕK ≤ JψK.


1. Since for a ∈ Ba ∨ ¬a = 1B

for every valuation ν we have that J(ϕ ∨ ¬ϕ)(ν)K = J(ϕ)(ν)K ∨ ¬ J(ϕ)(ν)K = 1B.

2. From the definition of boolean valued models, for τ ∈M :

Jτ = τK = 1B

hence for all valuations and variables J(x = x)(ν)K = 1B.

3. By definition we have

J∃ϕ(x)K =∨σ∈M

Jϕ(σ)K ≥ Jϕ(τ)K

hence we can conclude using Remark 2.3.13.

4. Using the definition of boolean valued model and of boolean evaluation of a formula, sup-pose ν(x) = τ and ν(y) = σ. We have then:

J(f(x) = f(y))(ν)K =∨ω∈M

(J(f(x) = ω)(ν)K ∧ J(f(y) = ω)(ν)K) =

∨ω∈M

∨χ∈Mn

Jτ = χK ∧ Jf(χ) = ωK

∧ ∨χ∈Mn

Jσ = χK ∧ Jf(χ) = ωK

≥∨ω∈M

((Jτ = σK ∧ Jf(σ) = ωK) ∧ (Jσ = τK ∧ Jf(τ) = ωK)) =

∨ω∈M

(Jτ = σK) ∧∨ω∈M

(Jf(σ) = ωK) ∧∨ω∈M

(Jf(τ) = ωK) = Jτ = σK ∧ 1B ∧ 1B = J(x = y)(ν)K

using the property ∨i∈I

(ai ∧ bi) =

(∨i∈I

ai

)∧

(∨i∈I

bi

)for ai, bi ∈ B. A proof of this fact can be found in [5, Corollary 2, Chapter 8].

5. It is sufficient to show that, for τ , σ ∈Mn:

Jτ = σK ≤ Jϕ(τ)→ ϕ(σ)K .

This is a consequence of Lemma 2.3.11, and of the fact that if a, b, c ∈ B are such thata ∧ b ≤ c, then a ≤ ¬b ∨ c. This follows from

a ≤ ¬b ∨ a = (¬b ∨ a) ∧ (¬b ∨ b) = ¬b ∨ (a ∧ b) ≤ ¬b ∨ c

6. It follows form a ≤ a ∨ b for all a, b ∈ B.

7. It follows from a = a ∨ a for all a ∈ B.

8. It follows from (a ∨ b) ∨ c = a ∨ (b ∨ c) for a, b, c ∈ B.

9. Let be JϕK = a, JψK = b, JχK = c. Then we have:

(a ∨ b) ∧ (¬a ∨ c) = (a ∧ ¬a) ∨ (a ∧ c) ∨ (b ∧ ¬a) ∨ (b ∧ c) ≤ c ∨ b ∨ (b ∨ c) = b ∨ c.


10. This item is proved as follows:

Jϕ(τ)→ ψK = J¬ϕ(τ) ∨ ψK ≤∨σ∈M

J¬ϕ(σ) ∨ ψK = J∃x(ϕ(x)→ ψ)K .

Remark 2.3.15. Since first order models are a subfamily of boolean valued models, we can inferwith no additional effort the Completeness Theorem also for boolean valued semantic.

Theorem 2.3.16 (Soundness and Completeness). Given a language L, an L-formula ϕ is syn-tactically provable from an L-theory T if and only if Jϕ(ν)K = 1B for all complete boolean algebrasB, all B-valued model M in which T is valid, and all valuations ν in M.

We now introduce quotients of B-valued models by a filter on B. Given a B-valued modelMin the language L and a filter F in B, we can define a B/F -valued model M/F whose domainwill be defined through an equivalence relation built on M using F . Some difficulties arise dueto the fact that B/F may not be necessarily complete, thus it might not be possible to satisfy allconditions in Definition 2.3.1 in order thatM/F satisfies Definition 2.3.8 for the boolean algebraB/F . Nonetheless completeness is not strictly necessary, since these definitions work once B/Fcontains “enough” suprema and infima.

Definition 2.3.17. Given B a boolean algebra and M a tuple defined as in 1-5 of Definition2.3.1, the couple 〈B,M〉 is a boolean couple in the language L if B contains all suprema andinfima required for Definitions 2.3.1 and 2.3.8.

We can generalize Definition 2.3.1 saying that M is a B-valued model if and only if 〈B,M〉is a boolean couple.

All the results we have presented so far for boolean valued model can be generalized toboolean couples. Nevertheless our analysis will focus on B valued models with B a completeboolean algebra.

Definition 2.3.18. Let 〈B,M〉 be a boolean couple in the language L and F a filter of B. Definethe equivalence relation on M

τ ≡F σ ⇔ Jτ = σK ∈ F

and M/F as the set of all equivalence classes [τ ]F : τ ∈ M. The B/F -model M/F =

〈M/F,=M/F , RM/Fi : i ∈ I, fM/F

j : j ∈ J, cM/Fk : k ∈ K〉 is defined as follows:

• =M/F is defined as:

=M/F : (M/F )2 → B/F

([τ ]F , [σ]F ) 7→[Jτ = σKMB

]F

• For any n-ary relation symbol R in L, let:

RM/F : (M/F )n → B/F

([τ1]F , . . . , [τn]F ) 7→[JR(τ1, . . . , τn)KMB

]F


• For any n-ary function symbol f in L, let:

fM/F : (M/F )n+1 → B/F

([τ1]F , . . . , [τn]F , [σ]F ) 7→[[Jfj(τ1, . . . , τn) = σKMB

]F

• For any constant symbol c in L, let cM/F = [cM]F ∈M/F .

Whenever 〈B/F,M/F 〉 is a boolean couple (except for vi, all points i-vii of Definition 2.3.1 arealways trivially satisfied), we say that the B/F -valued modelM/F is the quotient ofM by F .

Remark 2.3.19. The functions =M/F , RM/F , fM/F are all well-defined. For =M/F and RM/F

the proof is the same and we shall see in detail the case of =M/F . Let τ, τ ′, σ, σ′ ∈M such that[τ ]F = [τ ′]F and [σ]F = [σ′]F , we want to show

[Jτ = σK]F = [Jτ ′ = σ′K]F

Since both a = Jτ = τ ′K and b = Jσ = σ′K belong to F we have:

[a]F = 1B/F

[b]F = 1B/F

and since M is a B-valued model it follows that:

[Jτ = σK]F = [Jτ = σK]F ∧ [a]F ∧ [b]F = [Jτ = σK ∧ a ∧ b]F ≤ [Jτ ′ = σ′K]F .

In the same way it can be shown that [Jτ ′ = σ′K]F ≤ [Jτ = σK]F .For f an n-ary function symbol, the proof is similar but it requires Proposition 2.3.11: Let

(τ1, . . . , τn, σ) and (τ ′1, . . . , τ′n, σ

′) in Mn+1 such that a = Jτ = τ ′KMB and b = Jσ = σ′KMB are inF . We have

[Jf(τ1, . . . , τn) = σK]F = [Jf(τ1, . . . , τn) = σK]F ∧ [a]F ∧ [b]F =

[Jf(τ1, . . . , τn) = σK ∧ a ∧ b]F ≤ [Jf(τ ′1, . . . , τ′n) = σ′K]F

The other way around can be shown in the same way.

We show now that morphisms of boolean valued models are preserved by quotients.

Proposition 2.3.20. Let 〈B,M〉 and 〈C,N〉 be two boolean couples in the language L. Let F bea filter in B and i : B→ C an injective morphism of boolean algebras. Suppose both 〈B/F,M/F 〉and 〈C/i∗(F ),N/i∗(F )〉 are boolean couples, and Φ ⊆ M ×N is such that 〈i,Φ〉 is a morphismof boolean valued models. Let

ΦF = (α, β) ∈M/F ×N/i∗(F ) : ∃σ ∈ α, τ ∈ β such that (σ, τ) ∈ Φ.

Then 〈iF ,ΦF 〉 is a morphism between the boolean valued models M/F and N/i∗(F ). Moreover,if 〈i,Φ〉 is an injective morphism, embedding, or isomorphism of boolean valued models, then〈iF ,ΦF 〉 is respectively an injective morphism, embedding, or isomorphism of boolean valuedmodels.

Proof. Given (αj , βj) ∈ ΦF , we let σj ∈ M and τj ∈ N be two elements such that (σj , τj) ∈ Φand αj = [σj ]F , βj = [τj ]i∗(F ).

1. Since dom(Φ) = M , it follows that ΦF is everywhere defined.


2. Consider (α1, β1), (α2, β2) ∈ ΦF . We have then:

iF (Jα1 = α2K) = iF ([Jτ1 = τ2K]F ) = [i(Jτ1 = τ2K)]i∗(F ) ≤ [Jσ1 = σ2K]i∗(F ) = Jβ1 = β2K

3. Let R be an n-ary relation symbol in L and (α1, β1), . . . , (αn, βn) ∈ ΦF . We have that:

iF (JR(α1, . . . , αn)K) = iF ([JR(τ1, . . . , τn)K]F ) = [i(JR(τ1, . . . , τn)K)]i∗(F ) ≤

[JR(σ1, . . . , σn)K]i∗(F ) = JR(β1, . . . , βn)K

4. Consider f an n-ary function symbol in L and (α1, β1) . . . , (αn+1, βn+1) ∈ ΦF . Then:

iF (Jf(α1, . . . , αn) = αn+1K) = iF ([Jf(τ1, . . . , τn) = τn+1K]F ) =

[i(Jf(τ1, . . . , τn) = τn+1K)]i∗(F ) ≤ [Jf(σ1, . . . , σn) = σn+1K]i∗(F ) =

Jf(β1, . . . , βn) = βn+1K

5. Let c a constant symbol and (α, β) ∈ ΦF . Then:

iF (Jα = cK) = iF ([Jτ = cK]F ) = [i(Jτ = cK)]i∗(F ) ≤ [Jσ = cK]i∗(F ) = Jβ = cK

It can be easily checked that whenever equality holds in 2-5 of Definition 2.3.4, equality holdsas well in the respective points of this proof, and recalling Proposition 2.2.11, the proof isconcluded.

We also want to outline how some properties of a boolean valued model behave when passingto quotient models.

Definition 2.3.21. Given a boolean couple 〈B,M〉 we say thatM is an extensional B-valuedmodel if, given τ, σ ∈M :

τ = σ ⇔ Jτ = σK = 1B

Remark 2.3.22. If 〈i,Φ〉 is a morphism between the boolean couples 〈B,M〉 and 〈C,N〉 and Nis extensional, then Φ is a function, in fact given (τ, σ1), (τ, σ2) ∈ Φ:

1C = i(1B) = i(Jτ = τK) ≤ Jσ1 = σ2K

hence σ1 = σ2.

Proposition 2.3.23. Given a boolean couple 〈B,M〉 and a filter F in B, if 〈B/F,M/F 〉 is aboolean couple, then M/F is an extensional B/F -valued model.

Proof. Given τ, σ ∈M , we have that:

[τ ]F = [σ]F ⇔ Jτ = σK ∈ F

Recalling Remark 2.2.12, we can infer that:

[τ ]F = [σ]F ⇔ Jτ = σKB ∈ F ⇔ J[τ ]F = [σ]F KB/F = 1B/F


Example 2.3.24. We now introduce an example of extensional model inspired by functionalanalysis. Let L∞(R) be the set of all measurable functions with domain R. Let Meas be theboolean algebra of measurable subsets of R, and Null be the ideal on Meas given by the nullmeasure subsets of the real line. We shall denote with MALG the quotient boolean algebraMeas/Null. The family MALG with the operation of intersection, union, set complement, andwith set inclusion, is a complete boolean algebra (for more details on measure algebras see [5,Chapter 31]). We define a structure of MALG-valued model on L∞(R) in the language ≤,+, ∗.The definition of the boolean interpretations are the following (identifying any A ∈ Meas withits equivalence class in MALG):

Jf = gK = x ∈ R : f(x) = g(x)

Jf ≤ gK = x ∈ R : f(x) ≤ g(x)

Jf + g = hK = x ∈ R : f(x) + g(x) = h(x)

Jf ∗ g = hK = x ∈ R : f(x) ∗ g(x) = h(x)

It is immediate to check that L∞(R) is an MALG-valued model with these interpretations.Let F = R = 1M be the trivial filter on MALG, and consider the quotient M/F . Theboolean algebras MALG and MALG/F are clearly isomorphic. Hence MALG/F is complete, and〈MALG/F,L∞(R)/F 〉 is a boolean couple with L∞(R)/F an extensional MALG = MALG/F -model. In L∞(R)/F it holds that:

[f ]F = [g]F ⇔ Jf = gK ∈ F ⇔ Jf = gK = R

This means that we are identifying all functions that differ only on a null measure set. L∞(R)/Fis what is usually denoted with L∞(R).

Repeating verbatim the procedure above for any B-valued model M gives an extensionalmodel M/F with F = 1B. If M is already extensional, then M/F =M.

The following property is fundamental when considering the quotient of a boolean valuedmodel by a ultrafilter.

Definition 2.3.25. A B-valued modelM for the language L is full if for every L-formula ϕ(x, y)and every τ ∈M |y| there is a σ ∈M such that

J∃xϕ(x, τ)K = Jϕ(σ, τ)K

Remark 2.3.26. Consider M an extensional full B-valued model in the language L and f ann-ary function symbol in L. The boolean interpretation of f defines a function from Mn to M .We will denote it with the same name of the boolean interpretation of f , namely fM:

fM : Mn →M

(τ1, . . . , τn)→ σ

where σ is such that Jf(τ1, . . . , τn) = σK = 1B (here we use the hypothesis that M is full). Tosee this observe that whenever σ1, σ2 ∈M both satisfy the above property, then

1B = Jf(τ1, . . . , τn) = σ1K ∧ Jf(τ1, . . . , τn) = σ2K ≤ Jσ1 = σ2K

therefore, sinceM is extensional, it follows σ1 = σ2. When dealing with extensional full booleanvalued models, we will always refer to this function when considering the interpretation of afunction symbol in the language.


Theorem 2.3.27 (Boolean Valued Models Los’s Theorem). Assume M is a full B-valued modelfor the language L. Let G ∈ St(B). ThenM/G is a first order model for L and for every formulaϕ(x1, . . . , xn) in L and (τ1, . . . , τn) ∈Mn:

M/G |= ϕ([τ1]G, . . . [τn]G)⇔ Jϕ(τ1, . . . , τn)K ∈ G

Proof. As Remark 2.2.12 shows, B/G = [0B]G, [1B]G, so that 〈B/G,M/G〉 is always a booleancouple. Moreover ifM is full,M/G naturally defines a first order model (which will be denotedwith the same name). Given an n-ary relation symbol R, RM/G (as defined in Definition 2.3.18)is the characteristic function of the set (which we shall denote with the same name)

RM/G = ([τ1]G, . . . , [τn]G) ∈ (M/G)n : JR([τ1]G, . . . , [τn]G)K = 1B/G= ([τ1]G, . . . , [τn]G) ∈ (M/G)n : JR(τ1, . . . , τn)K ∈ G

Given f a function symbol, fM/G can be defined according to Remark 2.3.26, since M/G isextensional (see Proposition 2.3.23) and full (since M is full).

From this definition follows that

M/G |= f([τ1]G, . . . , [τn]G) = [σ]G ⇔ Jf(τ1, . . . , τn) = σK ∈ G

andM/G |= R([τ1]G, . . . , [τn]G)⇔ JR(τ1, . . . , τn)K ∈ G

We proof now the second part of the theorem. Denote with τ = (τ1, . . . , τn) a vector ofparameters in M and with [τ ]G = ([τ1]G, . . . , [τn]G) the vector of equivalence classes. With x, ywe denote variables and with c, c′ constants (and respectively with x and c the vectors). We willprove the Theorem just for a special class of formulae, those whose atomic subformulae are ofthe form

x = y c = y c = c′, (1)

R(x, c), (2)

f(x, c) = y f(x, c) = c. (3)

This is sufficient remarking that an atomic formula of the form

f(x) = g(y)

(or similarly defined using some constants as arguments for the functions) is logically equivalentto the formula

∃z(f(x) = z ∧ g(y) = z).

On the other hand an atomic formula with nested function symbols such as f(g(x)) = y islogically equivalent to the formula ∃z(g(x) = z ∧ f(z) = y). Thus by an easy induction we canprove that each formula is logically equivalent to one in which all the atomic subformale are ofthe form (1), (2) or (3). Thus it suffices to prove the theorem for this class of formulae:

Atomic formulae: Consider atomic formulae of the form

x = y c = y c = c′, (1)

R(x, c), (2)

f(x, c) = y f(x, c) = c. (3)

In this case the theorem easily follows from the preceding observations.


Negation: By induction, suppose the theorem holds for ϕ(x1, . . . , xn), then

M/G |= ¬ϕ([τ ]G)⇔M/G 6|= ϕ([τ ]G)⇔ Jϕ(τ)K /∈ G⇔ J¬ϕ(τ)K ∈ G

where last equivalence holds since J¬ϕ(τ)K = ¬ Jϕ(τ)K and G is a ultrafilter.

Conjunction: Suppose the Theorem holds for ψ(x1, . . . , xn) and for θ(x1, . . . , xn). Let

ϕ(x1, . . . , xn) ≡ ψ1(x1, . . . , xn) ∧ θ(x1, . . . , xn)

Then we have

M/G |= ϕ([τ ]G)⇔M/G |= ψ([τ ]G) and M/G |= θ([τ ]G)

⇔ Jψ(τ)K , Jθ(τ)K ∈ G⇔ Jψ(τ) ∧ θ(τ)K = Jψ(τ)K ∧ Jθ(τ)K ∈ G.

Where the last equivalence holds since G is a filter.

Existential: Let ϕ(x1, . . . , xn) = ∃yψ(x1, . . . , xn, y).

M/G |= ϕ([τ ]G)⇔M/G |= ψ([τ ]G, [σ]G) for some σ ∈M⇔ Jψ(τ , σ)K ∈ G for some σ ∈M⇔ J∃yψ(τ , y)K ∈ G.

where in the last equivalence ⇒ is always true, while the opposite direction holds sinceMis full.

Remark 2.3.28. Whenever L is a relational language (i.e. with no function symbols), in order tohave a first order model M/G when G is a ultrafilter in B, it is not necessary for M to be full.Nevertheless, without this request the previous theorem will not generally hold, as the followingexample will show.

Example 2.3.29. Consider B = RO(R) and Cω(R) the space of analytic functions from R to R,we will use this structure to produce a counterexample to the above Theorem. Let L = <,Cbe a relational language, where < is binary and C is unary. The boolean value of the relationsis defined as follows:

Jf = gK =˚x ∈ R : f(x) = g(x)

Jf < gK =˚x ∈ R : f(x) < g(x)

JC(f)K = x ∈ R : ∃I open interval such that x ∈ I and f I is constant

The set JC(f)K is always open, and it is regular for analytic functions (it is regular for continuousfunctions as well). Assume f is analytic on R such that it admits a non-empty interval on whichit is constant. Then f must be constant everywhere (see for example [10, Theorem 8.5,Chapter8]). Hence the only possibilities are JC(f)K = ∅ and JC(f)K = R for all functions f ∈ Cω(R).

With these definitions Cω(R) is a B-valued extension of the boolean couple 〈0, 1,R〉 (R isidentified with the constant functions cr(x) = r). Fix some f ∈ Cω(R) and consider the formula∃x(f < x ∧ C(x)), its boolean value can be calculated as follows:

J∃x(f < x ∧ C(x))K =∨g∈Cω

Jf < g ∧ C(g)K ≥∨r∈R

Jf < cr ∧ C(cr)K .


Since C(cr) = R, it follows that

Jf < cr ∧ C(cr)K = Jf < crK ∧ JC(cr)K = Jf < crK ;

therefore ∨r∈R

Jf < cr ∧ C(cr)K =∨r∈R

Jf < crK .

Consider now f (n,n+1) with n ∈ Z, and set mn = supx∈(n,n+1)(f(x)) + 1. It follows that:∨n∈Z

Jf < cmnK ≥∨n∈Z

(n, n+ 1) = R.

Hence J∃x(f < x ∧ C(x))K is a valid formula in the B-valued model considered.Now pick G ∈ St(B) extending H = (n,+∞) : n ∈ Z (such ultrafilter can be found since

H satisfies the finite intersection property, hence it generates a filter), and let N be the quotientCω(R)/G. We will show that

N |= [idR]G > y

for each y ∈ N such that N |= C(y). If C([g]G) holds in N , there is r ∈ R such that g = cr onR (since N |= C([g]G) if and only if JC(g)K = R). On the other hand we have that

JidR > crK = (r,+∞)

and this interval is in G since G contains (brc+ 1,+∞) ∈ H. This means that

N |= [idR] > [g]G

for all g such that JC(g)K ∈ G.In conclusion, even if ∃x(idR < x ∧ C(x)) is valid in Cω(R), this formula is not true in N .

We conclude this chapter with the following Lemma which will be needed later:

Lemma 2.3.30. Let M be a full B-valued model in the language L and ϕ an L-formula withparameters in M. For any b ∈ B the following are equivalent:

1. b ≤ JϕK,

2. there is D a dense subset of Ob such that for each G ∈ D M/G |= ϕ holds.

Proof. Assume b ≤ JϕK, the relevant implication immediately follows from Theorem 2.3.27 withD = Ob.

For the vice versa suppose b 6≤ JϕK. This implies that b∧¬ JϕK 6= 0B. From 0B < b∧¬ JϕK ≤ bit follows that Ob∧¬JϕK is a non-empty open subset of Ob. By Theorem 2.3.27 we have thatM/G |= ¬ϕ for any G ∈ Ob∧¬JϕK. In particular the set of G such that M/G |= ϕ is disjointfrom Ob∧¬JϕK and thus cannot be a dense subset of Ob.

33

Chapter 3

Set Theory and Forcing

We present a compact introduction to set theory and the basic properties of the forcing method.We assume the reader is already acquainted with the basic properties of set theory as formalizedin the first order axiomatization ZFC. Sections 1, 2 and 3 will be a brief summary of classicresults, references for those parts are [6], [8], [1], [3]. We will assume all over the chapter to workin a language L with just one binary relation symbol ∈.

3.1 Basics

Let V be the universe of sets, the standard model for set theory, i. e. the collection of all sets.

Definition 3.1.1. The cumulative hierarchy of sets is defined by recursion on the ordinals:

V0 = ∅

Vα+1 = P(Vα)

Vβ =⋃α<β

Vα if β is a limit ordinal

The Axiom of Regularity guarantees that every set belongs to some Vα (see [6, Lemma 6.3]),i. e.:

V =⋃

α∈ON

Vα

We define the rank of a set x as

ρ(x) = the least α such that x ∈ Vα+1

Remark 3.1.2. The universe V is not a set, thus we can not speak directly of it in ZFC, where onlyone type of objects are defined, namely sets. However, when dealing with ZFC, it is sometimesuseful to consider proper classes. In general, a class C is the extension in V of a certain firstorder formula C(x) in one or more free variables:

C = x : C(x)

Examples of classes we will consider are

ON = x : x is an ordinal

34 Chapter 3. Set Theory and Forcing

V = x : x = x

∈= (x, y) : x ∈ y

Classes are generally used in informal contexts as abbreviations. Given a class M and a binaryrelation E on it, we generalize the definition

〈M,E〉 |= ϕ

where ϕ is a ∈-formula. In the definition we gave of first order model we required the domainof the structure to be a set. With this assumption the relation M |= ϕ between sets M ∈ V and(Godel numbers for) formulae ϕ turns out to be a definable class in ZFC. However this is not thecase if we try to extend the relation |= so to include classes as the first argument of its domain.This issue can be solved showing that for each fixed ∈-formula ϕ and each pair of classes M andE ⊆ M2 the statement 〈M,E〉 |= ϕ is a definable class relation ϕM,E (in case E = ∈ we willsimply write ϕM ). Recall that ϕM,E is the formula obtained from ϕ substituting ∈ with E andall quantifiers ∃x and ∀x with ∃x ∈ M (i.e. ∃xM(x) ∧ . . .) and ∀x ∈ M (i.e. ∀xM(x) → . . .)respectively. Keeping this in mind we will informally generalize most of the notions defined forfirst order models. More about this in [8, Chapter 1, Section 9] and [6, Chapter 1, Chapter 12].

Chapters 6 and 12 of [6] contain all the details and the proofs we will omit in the remainingpart of this section.

Definition 3.1.3. A class A is transitive if x ∈ A implies x ⊂ A.

Definition 3.1.4. Consider a class P and a binary relation E on it. For each x ∈ P theextension of x is the class

extE(x) = z ∈ P : zEx

The relation E on P is:

• well-founded if every non-empty set A ⊂ P has an E-minimal element (i.e an x ∈ A suchthat zEx holds for no z ∈ A);

• set-like if extE(x) is a set for every x ∈ P ;

• extensional if given x, y ∈ P :

extE(x) 6= extE(y)⇒ x 6= y

or equivalently, if〈P,E〉 |= Axiom of Extensionality

Theorem 3.1.5 (Mostowski’s Collapsing Theorem). Assume E is a well-founded, set-like andextensional relation on a class P . Then there is a transitive class M and an isomorphism πbetween 〈P,E〉 and 〈M,∈〉. The transitive class M and the isomorphism π (the Mostowski’sCollapse of P ) are unique. Moreover, if E = ∈ and T ⊂ P is transitive, then π(x) = x for eachx ∈ T .

Proof. For a proof see, [6, Theorem 6.15].

Definition 3.1.6. Let 〈M,∈〉 and 〈N,∈〉 be two ∈-structures (possibly classes) such thatM ⊆ N . We say that a ∈-formula ϕ(x1, . . . , xn) is upward absolute for M,N if for every(a1, . . . , an) ∈Mn:

〈M,∈〉 |= ϕ(a1, . . . , an)⇒ 〈N,∈〉 |= ϕ(a1, . . . , an)

3.2. Boolean Valued Models for Set Theory 35

The formula ϕ(x1, . . . , xn) is downward absolute for M,N if for every (a1, . . . , an) ∈Mn:

〈M,∈〉 |= ϕ(a1, . . . , an)⇐ 〈N,∈〉 |= ϕ(a1, . . . , an)

A formula is absolute for M,N if it is both upward and downward absolute for M,N . A formulais (upward, downward) absolute for M if it is (upward, downward) absolute for M,V .

Theorem 3.1.7 (Reflection Principle). Consider a ∈-formula ϕ(x1, . . . , xn) and a set A.There exists an ordinal α such that:

• A ∈ Vα;

• ϕ is absolute for Vα.

In this case we say that Vα reflects ϕ.

Proof. For a proof see [6, Theorem 12.14].

3.2 Boolean Valued Models for Set Theory

The aim of this section is to formalize the boolean generalization of the universe V . V can bebuilt step by step iterating the operation of power set. Given a set X, P(X) can be identifiedwith the set of characteristic functions of its element, i. e. as the functions from X to theboolean algebra 0, 1. Roughly, the generalization of the power-set operation on a given set Xwe are looking for is obtained considering characteristic functions with domain X and range inan arbitrary complete boolean algebra instead of 0, 1. See Chapter 14 of [6] for a completeanalysis of this topic.

Definition 3.2.1. Consider a complete boolean algebra B. The class V B is defined by inductionon the ordinals as follows:

V B0 = ∅

V Bα+1 = f : X → B | X ⊂ V B

α

V Bβ =

⋃α<β

V Bα if β is a limit ordinal

V B =⋃

α∈ON

V Bα

The boolean rank of a B-name τ of V B is defined as:

ρB(τ) = the least α such that τ ∈ V Bα+1

At first, it is easier to define the structure of B-valued model on V B for L = ∈,⊆.

Definition 3.2.2. The interpretations of ∈,⊆ and = are defined by induction on the pairs(ρB(τ), ρB(σ)):

• Jτ ∈ σK =∨χ∈dom(σ)(Jτ = χK ∧ σ(χ));

• Jτ ⊆ σK =∧χ∈dom(τ)(τ(χ)→ Jχ ∈ σK);

• Jτ = σK = Jτ ⊆ σK ∧ Jσ ⊆ τK.


See [6, Lemmas 14.15 and 14.16] for a proof that the structure V B with the maps

(τ, σ) 7→ Jτ R σK

for R in =,∈,⊆ as defined above is a B-valued model for the language ∈,⊆.Remark 3.2.3. Definitions 3.2.1 and 3.2.2 work starting from any transitive first order modelM = 〈M,∈〉 of ZFC (the request for the model to be transitive is redundant, but we are mainlyinterested in transitive well-founded models in order to be able to build transitive generic exten-sions of such models). In this case, in order to have a boolean couple 〈B,MB〉 we need B ∈ Mand

M |= B is a complete boolean algebra

so that all subsets of B which belong to M admit a supremum in M itself. In this section wewill just analyze the boolean valued model V B, although our analysis can be easily generalizedso that it applies to any first order model of ZFC.

Remark 3.2.4. Every element x of V has a canonical name x in V B which we define by inductionon the rank:

• ∅ = ∅;

• for x ∈ V , x is the function whose domain is y : y ∈ x and such that x(y) = 1B for everyy ∈ x.

This map, when restricted to sets, is an embedding of B-valued models. In fact, given x, y ∈ V :

Jx ∈ yK =

1B if x ∈ y0B if x /∈ y

Jx ⊆ yK =

1B if x ⊆ y0B if x 6⊆ y

Jx = yK =

1B if x = y

0B if x 6= y

For a proof see [6, Lemma 14.21]

Theorem 3.2.5. Let B be a complete boolean algebra, then

• V B is full;

• ZFC is valid in V B.

Proof. See [6, Lemma 14.17, Lemma 14.19, Theorem 14.24].

Remark 3.2.6. In ZFC the binary relation x ⊆ y is introduced as an abbreviation for the formulain the language ∈:

y ⊆ x ≡ ∀z(z ∈ y → z ∈ x).

In order to show that V B is a B-valued model in L = ∈ we need to show that:

Jy ⊆ xK = J∀z(z ∈ y → z ∈ x)K

A proof of this is given in [1, Corollary 1.18].

Lastly we define a B-name which will be useful later.

Definition 3.2.7. GB ∈ V B is defined as:

dom(GB) = b : b ∈ B GB(b) = b

GB is the canonical name for a generic ultrafilter. If no confusion on the algebra Bconsidered can arise we shall omit the subscript B and denote GB by G.

3.3. Generic extensions 37

3.3 Generic extensions

From now on we will be interested in a structure 〈M,∈〉 (we will refer to it as M), which is atransitive first order model of ZFC (possibly a class), and to a B ∈M boolean algebra which Mmodels to be complete.

Definition 3.3.1. Given a partial order (P,<) we say that:

• D ⊆ P is dense if for every p ∈ P there is a d ∈ D such that d ≤ p;

• D ⊆ P is dense below q ∈ P if for every p ≤ q there is d ∈ D such that d ≤ p;

• D ⊆ P is open if p ∈ D and q ≤ p implies q ∈ D;

• D ⊆ P is predense if its downward closure

↓ D = b ∈ B : ∃d ∈ D(b ≤ d)

is dense.

We shall identify a boolean algebra B with the partial order B+ = B \ 0B and we generalizethe notion of filter to partially ordered sets as follows:

Definition 3.3.2. Given a partial order (P,<) a subset F ⊆ P is a filter if:

• F is non-empty;

• if p ≤ q and p ∈ F then q ∈ F ;

• if p, q ∈ F then there exists r ∈ F such that r ≤ p and r ≤ q.

Definition 3.3.3. Given a partial order (P,<), G ⊆ P is generic over a class C (or C-generic)if:

• G is a filter;

• if D ⊆ P is dense and D ∈ C, then G ∩D 6= ∅.

The following lemmas will be useful later. We will not omit the proof of the first one, as wewill need the map defined in it in the next chapter.

Lemma 3.3.4. Let B be a complete boolean algebra. There is a bijection between D, the familyof open dense subsets of B+, and E, the family of open dense subsets of St(B).

Ψ : E → DE 7→ a ∈ B+ : Oa ⊆ E

Proof. Well-defined: Assume E is open and dense in St(B). Consider b ∈ B+, it follows that

Ob ∩ E 6= ∅

We can therefore find a ∈ B+ such that

Oa ⊆ Ob ∩ E

Hence a ≤ b and a ∈ Ψ(E) hold. The set Ψ(E) is open since, given a, b ∈ B+

a ≤ b⇒ Oa ⊆ Ob


Injective: Let E 6= E′ be elements of E . We can assume there exists G ∈ E \ E′. E is open, wecan find therefore a ∈ B+ such that

G ∈ Oa ⊆ E

On the other hand we have:Oa 6⊆ E′

because G ∈ Oa. In conclusion, a ∈ Ψ(E) \Ψ(E′), hence the map is injective.

Surjective: Let D be a dense open subset of B+, we will show that

D = Ψ

(⋃a∈DOa

)

Consider b ∈ B+. There exists a ∈ D such that a ≤ b, so that

∅ 6= Oa ⊆ Ob

It follows that the intersection of⋃a∈DOa and Ob is non-empty. The thesis follows since

b ∈ B+ was arbitrary.

Lemma 3.3.5. Assume (P,<) is a partial order and D is a countable family of dense subsetsof P . Then there exists a D-generic filter on P . Moreover, for every p ∈ P there is a D-genericfilter on P containing p.

Proof. For a proof see, for example, [6, Lemma 14.4]

Definition 3.3.6. Let G be an M -generic ultrafilter over B. We define by induction on ρB theinterpretation by G of the B-names as:

• ∅G = ∅;

• τG = σG : τ(σ) ∈ G;

The generic extension of M by G is the class:

M [G] = τG : τ ∈MB

Remark 3.3.7. The previous definition can be given for any G ∈ St(B), and does not requireG ∈M . M [G] (which will not be called generic extension if G is not generic) is always transitive.We will always assume G to be M -generic for B unless otherwise stated.

Lemma 3.3.8. Assume G is an M -generic filter in B. Then for all τ, σ ∈MB:

τG ∈ σG ⇔ Jτ ∈ σK ∈ G;

τG ⊆ σG ⇔ Jτ ⊆ σK ∈ G;

τG = σG ⇔ Jτ = σK ∈ G.

Proof. For a proof see [6, Lemma 14.28]

Now we want to show which is the relationship between M [G] and MB/G. Since MB is aclass, given τ ∈ MB, [τ ]G might be a class itself. We can bypass this problem with the Scott’strick, defining [τ ]G as the set of σ ∈MB of minimal rank ρ such that σ ≡G τ .

3.3. Generic extensions 39

Theorem 3.3.9. Let M be a transitive model of ZFC, B a boolean algebra which M models tobe complete, and G an M -generic ultrafilter in B. The map

πMG : MB/G→M [G]

[τ ]G 7→ τG

is an isomorphism between the two models, more precisely it is the Mostowsky’s Collapse of the

extensional well founded model (MB/G,∈MB/G).

Proof. From Lemma 3.3.8 and Theorem 2.3.27 (〈MB,∈B〉 is a full boolean couple due to Theorem3.2.5) it follows that πMG is an L-embedding. The map is surjective, since for any x ∈M [G], bydefinition x = τG for some τ ∈MB, hence

x = πMG ([τ ]G)

Theorem 3.3.10 (Cohen’s Forcing Theorem). Assume M is a transitive first order model ofZFC, B a boolean algebra which M models to be complete and G an M -generic filter in B. Letϕ(x1, . . . , xn) be an L-formula, then for every τ1, . . . , τn ∈MB:

M [G] |= ϕ(τG1 , . . . , τGn )⇔ Jϕ(τ1, . . . , τn)K ∈ G

Proof. The map πMG defined in Theorem 3.3.9 is an isomorphism between MB/G and M [G],therefore

M [G] |= ϕ(τG1 , . . . , τGn )⇔MB/G |= ϕ([τ1]G, . . . , [τn]G)

MB is full, therefore we can conclude the proof using Theorem 2.3.27.

The last theorem we present in this section tells us that M [G] is the smallest transitive firstorder model N such that M ⊂ N and G ∈ N .

Theorem 3.3.11 (Generic Model Theorem). Let M be a transitive model of ZFC, B a booleanalgebra which M models to be complete and G an M -generic ultrafilter in B. Then 〈M [G],∈〉 isa first order transitive model such that:

• M [G] |= ZFC;

• M ⊆M [G] and G ∈M [G];

• ONM = ONM [G];

• if N is a transitive first order model of ZFC such that M ⊆ N and G ∈ N then M [G] ⊆ N .

Proof. To prove the first item we observe that by Theorem 3.2.5 all axioms of ZFC have booleanvalue equal to 1B, which clearly belongs to any ultrafilter in St(B). The thesis follows thereforefrom Theorem 3.3.10.

The second item holds since it can be shown that or each x ∈M

xG = x

and also thatGG = G

A detailed proof of this and of the other two items can be found in [6, Lemma 14.31].


3.4 More on Cohen’s Forcing Theorem

The hypotheses of Cohen’s Forcing Theorem are too strict, and on a first sight they make thetheorem useless: on the one hand we assumed the existence of a transitive first order model ofZFC M , which is unprovable within ZFC because of Godel Incompleteness Theorem. On theother hand, we assumed the existence of an M -generic ultrafilter. Unfortunately, the existenceof such an ultrafilter can not generally be proved, unless M satisfies certain strict requirements,for example M -generic ultrafilters extists whenever M is a countable model (see Lemma 3.3.5).Therefore, one may ask how we can use Cohen’s Forcing Theorem in order to show that a certainformula ϕ is consistent with ZFC. In this section we provide a method to bypass the problemsexposed above. The first observation is that, when proving that ϕ is consistent with ZFC bymeans of Cohen’s Forcing Theorem, by a compactness argument, we only need to prove that ϕ isconsistent with an arbitrary finite set of ZFC-axioms. We can also argue that in order to show thatin an M -generic extension M [G] a finite set of ZFC-axioms hold, we just need M to be a modelof a (possibly different) finite set of ZFC-axioms. Finally, we can use the Reflection Principle3.1.7, the Downward Lowenheim−Skolem Theorem 2.1.20, and the Mostowski’s Collapse 3.1.5to find countable transitive models M of any given finite set of ZFC-axioms. Such models M willbe the ones to which we can apply Cohen’s Forcing Theorem.

We will need to work among different classes which may or may not be models of ZFC, hencewe introduce some definitions regarding how a formula behaves when we consider it in differentmodels

The behaviour of formulae between different structures is mainly determined by their logicalcomplexity. A precise formalization of this concept can be given by means of the Levy Hierarchy(see [6, Chapter 13]). For our purposes the following definitions are sufficient.

Definition 3.4.1. A ∈ ∪ a1, . . . , an-formula ϕ(x1, . . . , xn, a1, . . . , an) (with a1, . . . , an addi-tional constant symbols) is ∆0 if

• it has no quantifiers;

• it is of the form ϕ ∧ ψ, ϕ ∨ ψ, ¬ϕ, ϕ→ ψ, ϕ↔ ψ and both ϕ and ψ are ∆0;

• it is of the form(∃x ∈ y)ψ(x, x1, . . . , xn, a1, . . . , an)

or(∀x ∈ y)ψ(x, x1, . . . , xn, a1, . . . , an)

and ψ is1 ∆0.

We will loosely say that ϕ is ∆0 over a theory T if it is provable from T that ϕ is equivalent toa formula of the form above. A formula ϕ is Σ1 (Π1) in T if it is provable to be equivalent inT to one of the form ∃xψ(x) (∀xψ(x)) where ψ is a ∆0-formula. A formula ϕ is ∆1 in T if it isboth Π1 and Σ1 in T . If not otherwise stated we will assume T = ZFC and we will just say thatϕ is ∆0 (or Σ1,Π1,∆1).

Proposition 3.4.2. Given a transitive ∈-structure M , every ∆0-formula is absolute for M .

Proof. For a proof see [6, Lemma 12.9]

Remark 3.4.3. An easy consequence of this is that, when considering transitive classes, Σ1-formulae are upward absolute, Π1-formulae are downward absolute, and ∆1 formulae are absolute.

1Recall that (∃x ∈ y)ψ(x) is a shorthand for ∃x(x ∈ y ∧ ψ(x)) and similarly (∀x ∈ y)ψ(x) is a shorthand for∀x(x ∈ y → ψ(x)).

3.4. More on Cohen’s Forcing Theorem 41

Consider a complete boolean algebra B. We need to generalize the definition of V B to transi-tive classes M which are not necessarily models of ZFC and are such that B ∈M . We will performthis generalization studying the formula which defines V B, although this is not the quickest wayto do it. Notice that the properties which define a boolean algebra can always be expressed withquantifiers bounded in B. Hence “B is a boolean algebra” and its operations and relations areabsolute for transitive classes. Moreover, B ∈M and M is transitive imply B ⊂M , so that if Bis complete in V , it has to be complete in M as well (every supremum which is in V is also inM). Observe that the vice versa does not hold. In fact, if B is a complete boolean algebra in atransitive class M , and N is a transitive class containing M , there might exists some subset ofB belonging to N but not to M , thus that set may not have a supremum.

Definition 3.4.4. Given a set X, the transitive closure of X is defined as:

TC(X) =⋃(⋃

X)n

: n ∈ ω

i.e. the smallest transitive set containing X.

Definition 3.4.5. Given a complete boolean algebra B and f : V → B a partial function (i. e.whose domain is contained in V ) which is an element of V , we define:

∗⋃f =

⋃dom(x) : ∃b(x, b) ∈ f ∧ x is a partial function with range in B

and we define

TCB(f) =⋃( ∗⋃

f

)n: n ∈ ω

where ( ∗⋃

f

)0

= dom(f)

( ∗⋃f

)1

=

∗⋃f

( ∗⋃f

)n+1

=

∗⋃( ∗⋃f

)nfor n ≥ 1

We call TCB(f) the boolean transitive closure of f on B.

Lemma 3.4.6. Let B be a complete boolean algebra, then τ ∈ V B if and only if

Φ(τ,B) ≡ τ is a function ∧ Im(τ) ⊆ B ∧ ∀σ ∈ TCB(τ)(σ is a function ∧ Im(σ) ⊆ B)

Moreover Φ(x,B) is ∆1 in the parameter B.

Proof. We first check the above equivalence. Let τ be a B-name, hence it is a function withrange in B. Then we proceed by induction: if σ ∈ dom(τ) then by definition σ ∈ V B hence it is

a function whose rank is contained in B. Consider now σ ∈( ∗⋃

τ

)n+1

. The element σ is in the

domain of some η ∈( ∗⋃

τ

)n, which by inductive hypothesis is in V B, hence σ is a B-name.


Conversely, towards a contradiction let τ be of minimum rank such that Φ(τ) holds butτ /∈ V B. Since

∀σ ∈ TCB(τ)(σ is a function ∧ Im(σ) ⊆ B)

holds, every η ∈ dom(τ) is a function whose image is in B. Moreover

∀σ ∈ TCB(η)(σ is a function ∧ Im(σ) ⊆ B)

hence Φ(η,B) holds. Since x ∈ y implies ρ(x) < ρ(y) and for a certain b ∈ B

η ∈ η ∈ η, η, b = (η, b) ∈ τ

we have ρ(η) < ρ(τ). Thus η ∈ V B and from this follows that τ is a function in B whose domainis composed by B-names, hence it is a B-name itself.

For the proof that Φ is ∆1 we will assume most of the basic facts about the Levy Hierarchy.For those results see [3, Lemma 2.6, Theorem 2.7, Lemma 2.8] and [6, Lemma 12.10]. This meansthat we only need to show that y = TCB(f) is ∆1 with the parameter B. In order to do this, it

is enough to prove that x ∈∗⋃f is ∆1, since

y = TCB(f)↔ ∀x ∈ y∃z ∈ ω(x ∈ (

∗⋃f)z)

and the fact that x ∈ (∗⋃f)n is ∆1 easily follows once we have showed that x ∈

∗⋃f is ∆1 (trivial

for n = 0, otherwise just substitute f with (∗⋃f)n−1). On the one hand we have

x ∈∗⋃f ↔ ∃z∃b ∈ B((z, b) ∈ f ∧ z is a function ∧ Im(z) ⊆ B ∧ x ∈ dom(z))

and the right side formula is Σ1. On the other hand

y ⊇∗⋃f ↔ ∀z [(z is a function ∧ Im(z) ⊆ B ∧ ∃b ∈ B((z, b) ∈ f))→ dom(z) ⊆ y]

tells us that y ⊇∗⋃f is Π1, and since

x ∈∗⋃f ↔ ∀y(y ⊇

∗⋃f → x ∈ y)

we have the thesis.

Remark 3.4.7. The B-names in a transitive class M are defined as those elements τ such thatΦ(τ,B)M holds. With the previous lemma we have showed that Φ(x,B) is absolute for transitivestructures, hence if τ ∈M

Φ(τ,B)M ⇔ Φ(τ,B)

so that the class of B-names in M , which we shall call MB, overlaps with V B ∩M . We need tocheck that MB inherits the structure of B-valued model.

Proposition 3.4.8. Let M be a transitive class and B complete boolean algebra which belongsto M . Then MB = V B ∩M with relations defined for σ, τ ∈MB by:

Jσ ∈ τKMB

= (Jσ ∈ τKVB

)M

3.4. More on Cohen’s Forcing Theorem 43

Jσ = τKMB

= (Jσ = τKVB

)M

is a B-valued model for the language ∈. Moreover if ϕ(x1, . . . , xn) is a formula and τ1, . . . , τn ∈MB then:

Jϕ(τ1, . . . , τn)KMB

= (Jϕ(τ1, . . . , τn)KVB

)M

Proof. For the first part of the proof it suffices to show that Jσ ∈ τKVB

and Jσ = τKVB

are absolutefor transitive models M such that B ∈ M and M models B to be a complete boolean algebra.We will only prove the ∈-clause, the one with ⊆ (from which the one with = follows) can beinferred similarly. The proof proceeds by induction on the pairs (ρB(σ), ρB(τ)). For the emptyset:

(J∅ ∈ ∅KVB

)M = (0B)M = 0B = J∅ ∈ ∅KVB

.

Given σ, τ ∈MB:

(Jσ ∈ τKVB

)M =∨

η∈dom(τ)∩M

(Jη = σKVB

)M ∧ τ(η).

By the inductive hypothesis (Jη = σKVB

)M = Jη = σKVB

, and since τ ∈ M , we have τ ⊆ M , sothat if η ∈ dom(τ) then (η, b) ∈M for some b ∈ B. By the definition of ordered couple

(η, b) = η, η, b

and from the transitivity of M it follows that η ∈ M , hence dom(τ) ⊆ M , which leads to thethesis.

The second part of the proposition is by induction on the complexity of ϕ. The case for ϕ anatomic formula holds by definition. The proof for conjunction and negation is immediate, lastlyfor ϕ ≡ ∃xψ(x) we have that:

(JϕKVB

)M =∨

τ∈V B∩M

(Jψ(τ)KVB

)M =∨

τ∈MB

Jψ(τ)KMB

= JϕKMB

.

Remark 3.4.9. Notice that defining Jσ ∈ τKMB

and Jσ = τKMB

as in the proposition or as inDefinition 3.2.2 gives the same result.

We have defined MB for any transitive class M which contains B. Since ZFC does notnecessarily hold in M , we do not know which properties of V B (fullness, validity of ZFC, etc.)hold in MB.

Definition 3.4.10. Given a Boolean algebra B, A ⊆ B\0B is an antichain if any two distinctx, y ∈ A, are incompatible, i. e. x ∧ y = 0B.

The following Lemma is crucial if we are looking for a full B-valued model, it holds for everymodel of ZFC (see Lemma 14.18 in [6]), and it implies that V B is full.

Lemma 3.4.11 (Mixing Lemma for B). Assume B is a complete boolean algebra in V . Let A bean antichain of B and for any a ∈ A let τa be an element of V B. Then there exists some τ ∈ V B

such that a ≤ Jτ = τaK for all a ∈ A.

In [6, Lemma 14.19] is shown how the Mixing Lemma for B implies that V B is full. In thisproof some axioms of ZFC are used (Zorn’s Lemma, the axiom schema of separation and ofreplacement). We are not interested in a full analysis of which of them are necessary and whichredundant. For our purposes consider the following:


Proposition 3.4.12. Let M be a transitive class which is a model of the axiom schema ofseparation, of replacement, and of the axiom of choice. Let moreover B ∈ M a boolean algebrawhich M models to be complete. If:

M |= Mixing Lemma

then MB is full.

Proof. The proof is as in [6, Lemma 14.19].

Observe that in order to prove the proposition above for a single formula ϕ, i. e. that there

exists τ ∈ V B such that J∃xϕ(x)KMB

= Jϕ(τ)KMB

, by compactness only a finite number of axiomsof ZFC are needed.

Remark 3.4.13. So far we have defined the class MB for any transitive M such that B ∈ Mand M models B to be a complete boolean algebra. Given an M -generic filter G in B, we cangeneralize to these models also Definition 3.3.6, i. e. the generic extension M [G]. We do notknow if M models ZFC, hence we can not infer Theorems 3.3.9 and 3.3.10 for M [G]. However ifwe want these theorems to hold for a specific formula ϕ in M [G], only a finite number of axiomsof ZFC is required to hold in M . For instance we can require that Lemma 3.3.8 holds in M , andwe can ask MB to be full for ϕ (which can be obtained through specific instances of the MixingLemma as described above). By compactness, given a formula ϕ and a transitive class M with acomplete boolean algebra B ∈M , by requiring that M satisfies a certain formula Θϕ we can inferthat Theorem 3.3.9 holds in M whenever G is M -generic, and that Cohen’s Forcing Theorem3.3.10 holds in M for ϕ. Θϕ will depend on which instance ϕ of Cohen’s Forcing Theorem wewant to satisfy.

Now we can finally present the final result of this section. We state it for V a transitive ZFC-model. However with some more intricacies in its formulation, it can be inferred for arbitraryfirst order models of ZFC.

Proposition 3.4.14. Let B be a complete boolean algebra in V . Let b ∈ B, consider a formulaϕ(x1, . . . , xn) and τ1, . . . , τn ∈ V B. Let α > ω be an ordinal (given by the Reflection Principle)

such that τ1, . . . , τn, b,B ∈ Vα and such that Vα reflects (Jϕ(τ1, . . . , τn)KVB

≥ b) ∧ Θϕ. Then thefollowing are equivalent:

1. Jϕ(τ1, . . . , τn)KVB

≥ b.

2. For every M ∈ V such that M ≺ Vα and M is a countable structure to which τ1, . . . , τn, b,Bbelong, if π : M → N is the Mostowski Collapse, and G ⊆ π(B) is an N -generic ultrafiltersuch that π(b) ∈ G, then N [G] |= ϕ(π(τ1)G, . . . , π(τn)G).

3. There exists some countable M ≺ Vα with M ∈ V and τ1, . . . , τn, b,B ∈M such that lettingπ : M → N be the Mostowski Collapse, and G ⊆ π(B) be any N -generic ultrafilter suchthat π(b) ∈ G, then N [G] |= ϕ(π(τ1)G, . . . , π(τn)G).

Proof. 1 ⇔ 2. Fix M ≺ Vα. Since being a B-name is absolute for transitive models andπ is an isomorphism, we have that π(τ1), . . . , π(τn) ∈ Nπ(B). Moreover, since Vα reflects

Jϕ(τ1, . . . , τn)KVB

≥ b, M is an elementary substructure of Vα and π is an isomorphism, wecan infer that:

Jϕ(τ1, . . . , τn)KVB

≥ b⇔ (Jϕ(τ1, . . . , τn)KVB

≥ b)Vα

⇔ (Jϕ(τ1, . . . , τn)KVB

≥ b)M

⇔ Jϕ(π(τ1), . . . , π(τn))KNπ(B)

≥ π(b).

3.5. Absoluteness results 45

For the same reasons Θϕ holds in N . Moreover, Lemma 3.3.5 tells us that N -generic ultrafilterare dense in St(π(B)). Then one direction of the thesis follows from Theorem 3.3.10, the otherfrom Lemma 2.3.30.

1⇔ 3. The existence of a countable model M such that M ≺ Vα is guaranteed by DownwardLowenheim−Skolem Theorem 2.1.20. With this observation the proof proceeds as in the previouspoint.

Remark 3.4.15. Let ϕ be a formula. Suppose that ZFC proves that for some boolean algebraB JϕKB > 0B. By compactness only a finite number of axioms of ZFC are needed to prove thisderivation. We let Ξϕ denote the conjunction of these axioms. By the Reflection Principle 3.1.7we get a Vα which reflects Ξϕ ∧ Θϕ. By the Downward Lowenheim−Skolem Theorem 2.1.20,and the Mostowski’s Collapse 3.1.5, we can find a countable transitive set N which is a modelof Ξϕ ∧ Θϕ. Since N is a countable model of Ξϕ, there exists a boolean algebra B ∈ N whichN models to be complete and such that JϕKB > 0B. By Lemma 3.3.5 there exists an N -genericultrafilter G with JϕKB ∈ G. We conclude, by means of Cohen’s Forcing Theorem applied to ϕ,that N [G] |= ϕ whenever G is N -generic for B with JϕKB ∈ G (this is the case since N modelsΘϕ).

Since we can repeat this procedure for any finite set of axioms of ZFC which includes Ξϕ∧Θϕ,we obtain that ϕ is finitely consistent with ZFC, and thus, by compactness, that it is consistentwith ZFC.

This explains why set theorists can use Cohen’s Forcing Theorem without being worried bythe non-existence of a V -generic ultrafilter.

3.5 Absoluteness results

The previous sections provided the basic tools needed in order to obtain consistency results usingforcing. Now we will present some theorems which show how the forcing techniques can be used toderive properties within ZFC. The key of this process will be Shoenfield’s Absoluteness Theorem.An exhaustive description of this topic can be found in Chapter 25 of [6]. Our description willbe slightly different, and it will be closer to the introduction of [13].

Definition 3.5.1. Let κ be a cardinal. We define the set Hκ as

Hκ = x : |TC(x)| < κ

Proposition 3.5.2. Assume κ > ω is regular. Then Hκ is a model of all axioms of ZFC exceptfor the power set axiom.

Proof. See [8, Theorem 6.5]

Theorem 3.5.3 (Cohen’s Generic Absoluteness). Assume ϕ(x, a) is a ∆0-formula and a ⊆ ω.The following are equivalent:

1. 〈Hℵ1 ,∈〉 |= ∃xϕ(x, a);

2. there is a complete boolean algebra B such that J∃xϕ(x, a)KVB

> 0B.

Proof. Suppose 〈Hℵ1 ,∈〉 |= ∃xϕ(x, a), since Hℵ1 ⊂ V and Σ1-formulae are upward absolute itholds

V |= ∃xϕ(x, a)

hence, considering B = 0, 1, we have J∃xϕ(x, a)KVB

= 1B.


Conversely, assume there is a complete boolean algebra B such that J∃xϕ(x, a)KVB

> 0B. ByRemark 3.4.15 there exists a countable transitive model N and an N -generic ultrafilter G suchthat N [G] |= ϕ(s, a) for a certain s ∈ N [G] (notice that π(a) = a since a ⊆ ω). The set N [G] istransitive and countable, hence it is contained in Hℵ1 and so does a. The formula ϕ is ∆0, soit is absolute for transitive models, hence Hℵ1 |= ϕ(s, a) (every Hκ is clearly transitive), whichleads to Hℵ1 |= ∃xϕ(x, a).

Remark 3.5.4. Cohen’s Absoluteness Theorem deserves some comments. We know that (Hℵ1)V ⊆V for any V |= ZFC, and Σ1-formulae are upward absolute. Thus Cohen’s Absoluteness Theoremtells us that in order to proof ∃xϕ(x, a) (ϕ any ∆0-formula) in ZFC, it is not necessary to showthat this formula holds in every model of ZFC, but just that J∃xϕ(x, a)K > 0B holds for some B.This can be performed finding one appropriate model through forcing. This technique can alsobe applied to Σ1

2-formulae, which are a specific class of formulae in the second order arithmetic(see [6, Chapter 25] for more details), since these formulae can be translated into Σ1-formulae inHℵ1 (see Lemma 25.25 in [6]).

49

Chapter 4

C∗-algebras and B-names forcomplex numbers

In this chapter we will finally draw a bridge between the theory of commutative C∗-algebras andthe theory of boolean valued models of set theory. First we will show how to define (given B acomplete boolean algebra) a structure of B-valued extension of C on the C∗-algebra C(St(B)),the set of continuous functions from St(B) to C. In the second part we will consider C∗-algebrasA whose spectrum is extremely disconnected, and we will show that

A ∼= C(St(B))

for a specific complete boolean algebra B. In the last section we will provide an embedding ofthese C∗-algebras in the B-names for complex numbers of V B. From this we will get that thequotient C(St(B))/G, where G is a V -generic ultrafilter of B, is an algebraically closed field whichextends C, and which preserves the truth value of Σ2-formulae of C.

4.1 A boolean valued extension of CWe introduce some definitions from topology.

Definition 4.1.1. Let X be a topological space.

• A ⊆ X is nowhere dense if its complement contains an open dense set;

• A ⊆ X is meager if it is the union of countably many nowhere dense sets;

• A ⊆ X is comeager if its complement is meager;

• A ⊆ X has the Baire property if there exists an open set U ⊆ X such that A∆U ismeager.

The family of Borel sets of X is the σ-algebra generated by the open subsets of X, i. e. thesmallest family F of subsets of X such that:

1. if A ⊆ X is open then A ∈ F ;

2. if A ∈ F then Ac ∈ F ;

50 Chapter 4. C∗-algebras and B-names for complex numbers

3. if Ann∈ω ⊆ F then⋃n∈ω An ∈ F .

We need a couple of properties before being able to present the boolean extension of C.

Proposition 4.1.2. Assume X is a compact Hausdorff space. Then every Borel set B of X hasthe Baire property and there exists a unique regular open set U such that B∆U is meager.

Proof. For a proof see [5, Chapter 29, Lemma 2].

Proposition 4.1.3. Let X,Y be two topological spaces and f : X → Y a continuous map. IfB ⊆ Y is a Borel set, then f−1[B] is a Borel set of X.

Proof. This is the case since the preimage of open sets through f is open, and:

f−1

[⋃n∈ω

An

]=⋃n∈ω

f−1[An]

f−1[Acn] = f−1[An]c

The following example shows how to obtain a boolean extension of a topological space Xwhen the language is composed by symbols which are interpreted as Borel subsets of Xn.

Example 4.1.4. Fix a complete boolean algebra B, a topological space X, and R ⊆ X × Xa binary Borel relation on X. Consider M = C(St(B), X) the set of continuous functions fromSt(B) to X. We can define a structure of B-valued extension of X on M . Given f, g ∈ M , theset

W = G ∈ St(B) : f(G)Rg(G) = (f × g)−1(R)

is a Borel subset of St(B) since both f and g are continuous. By Proposition 4.1.2 W has theBaire property and

˚G ∈ St(B) : f(G)Rg(G)

is the unique regular open set U ⊆ St(B) such that W∆U is meager. If we identify B andRO(St(B)) (B is complete), we have that

RM (f, g) =˚G ∈ St(B) : f(G)Rg(G)

is a well defined element of B. We can repeat verbatim the procedure above in order to definethe boolean interpretation of any n-ary Borel relation on X and of any function

F : Xn → X

whose graph is a Borel subset of Xn+1 (we will simply say that F is a Borel function). Theboolean interpretation of the equality can be defined as long as the set

∆X = (x, x) ∈ X ×X : x ∈ X

is a Borel set in X ×X. With these definitions it can be checked that M is a B-valued model.Moreover the set cx ∈M : x ∈ X, where cx is the constant function with value x, is a copy ofX in M in the sense that, for x, y ∈ X:

xRy ⇔ JcxRcyK = 1B

4.2. C∗-algebras as B-valued models 51

¬xRy ⇔ JcxRcyK = 0B

x = y ⇔ Jcx = cyK = 1B

x 6= y ⇔ Jcx = cyK = 0B

Thus we can infer that M is the boolean extension of an isomorphic copy of X seen as a 2-valuedmodel.

Remark 4.1.5. If X = C we have that ∆C is closed (C is Hausdorff). Hence, fixed a language Lwhose elements are Borel relations and functions in C, we can define in the C∗-algebra C(St(B))a structure of B-valued extension of C for the language L.

4.2 C∗-algebras as B-valued models

Definition 4.2.1. Let X be a 0-dimensional compact Hausdorff space. Consider the function

ϕX : X → St(CL(X))

x 7→ Gx

with Gx = A ∈ CL(X) : x ∈ A the ultrafilter defined in Proposition 2.2.17. We shall call ψXthe inverse map of ϕX and, for G ∈ St(B)

ψX(G) = xG

We recall that xG =⋂G.

Proposition 4.2.2. Let A be a commutative unital C∗-algebra, X = σ(A) its spectrum andB = RO(X). Assume furthermore that X is extremely disconnected. Then X is homeomorphicto St(B) and the map

ΦA : C(X)→ C(St(B))

f 7→ f

where f(G) = f(xG), is an isometric ∗-isomorphism of C∗-algebras.

Proof. Proposition 1.2.8 implies that X is regular, hence it admits a basis of regular open sets,and since RO(X) = CL(X) we have that X is 0-dimensional. By Proposition 2.2.17, and the factthat B = CL(X), we infer that X ∼= St(CL(X)) = St(B). Thus the map


f 7→ f

is an homeomorphism. By the definition of f it easily follows that ΦA is an isometric ∗-isomorphism.

Remark 4.2.3. The previous proposition connects abelian unital C∗-algebras whose spectrum isextremely disconnected with boolean extensions of the complex field. In fact, assume that A isa C∗-algebra with such properties, X its spectrum and B = RO(X). If we combine the previousresult with the Gelfand-Naimark Theorem 1.2.10, we obtain :

A ∼= C(X) ∼= C(St(B))


Now we want to see what happens when we consider a commutative unital C∗-algebra whosespectrum is not extremely disconnected. In order to do this we will generalize the map ΦAdefined in Proposition 4.2.2. First we need to generalize Definition 4.2.1.

Lemma 4.2.4. Let X be a compact Hausdorff topological space and B = RO(X). The map

ψX : St(B)→ X

G 7→ xG

where xG is the only element in

CG =⋂U : U ∈ G

is well-defined, continuous and surjective. This map is the function ψX given by Definition 4.2.1if and only if X is extremely disconnected.

Proof. Well-defined: We have to show that, given G ∈ St(B), CG is a singleton. The set isnon-empty since CG inherits the finite intersection property from G, and X is compact.The space X is regular, hence if x 6= y ∈ CG there exists a regular open set A such thatx ∈ A and y /∈ A. From this and

A ∈ G⇒ ˚(X \A) /∈ G

follows thatx ∈ CG ⇒ y /∈ CG

which is absurd.

Surjective: For each x ∈ X consider a ultrafilter G which extends the filter of the regular opennieghborhoods of the point. In this case xG = x.

Continuous: Let A be an open set in X, and let G be such that xG ∈ A. Let B a regular openset such that xG ∈ B and B ⊆ A. B is open and xG /∈ Bc, so that xG /∈ ¬B. This meansthat B ∈ G (otherwise ¬B ∈ G, which is absurd since xG 6∈ ¬B), and from B ⊆ A followsOB ⊆ ψ−1(A).

The following theorem generalizes Proposition 4.2.2 to any commutative unital C∗-algebra.

Theorem 4.2.5. Fix a commutative unital C∗-algebra A, let X be its spectrum, and let B =RO(X). The function


f 7→ f

where f(G) = f(xG), is an injective ∗-homomorphism of C∗-algebras. Moreover, the map ΦA isan isometric ∗-isomorphism if and only if X is extremely disconnected.

Proof. The map clearly preserves sum, product and involution.

Continuous: These inequalities guarantee continuity:

‖ΦA(f)‖ = maxG∈St(B)

∣∣∣f(G)∣∣∣ = max

G∈St(B)|f(xG)| ≤ ‖f‖

4.2. C∗-algebras as B-valued models 53

Injective: Consider f 6= 0, hence there exists x ∈ X such that f(x) 6= 0. From Lemma 4.2.4 we

know that there exists G ∈ St(B) such that x = xG, thus f(G) 6= 0.

Surjective: As we have showed earlier, if X is extremely disconnected the map ΦA is an isometric∗-isomorphism. Suppose X is not extremely disconnected and consider the surjective con-tinuous map ψX in Lemma 4.2.4. Since both X and St(B) are compact Hausdorff spaces,the map can not be bijective, otherwise it would be an homeomorphism, which is impossi-ble since St(B) is extremely disconnected. Thus injectivity of ψX must fail, and there exist

two ultrafilters G and H such that xG = xH . We conclude that for every f ∈ Im(ΦA)

f(G) = f(H)

(even more, f is constant over the fiber of ψX at any point x ∈ X). Since St(B) is a normalspace we can always find an h ∈ C(St(B)) which assumes different values on different pointswhich are on the same fiber of ψX . We conclude that h can not belong to Im(ΦA).

Remark 4.2.6. Summing up, given A a unital commutative C∗-algebra and B = RO(σ(A)), themap ΦA ΓA is a ∗-homomorphism which embeds A in C(St(B)), a B-valued extension of C. Wehave also showed that this map is a ∗-isomorphism if and only if σ(A) is extremely disconnected.We may ask whether this request is too strong and if only exotic examples of C∗-algebras satisfyit. This is not the case, and we will provide an example.

Example 4.2.7. Let A = L∞(C) and X be its spectrum. Working directly with B = RO(X)is a bit laborious. We consider then MALG, the complete boolean algebra of measurable setsin C modulo null measure sets. We will work in this more familiar context, and show thatX ∼= St(MALG). This will be enough, in fact this will guarantee thatX is extremely disconnected,hence X ∼= St(RO(X)), and by Corollary 2.2.20 (which implies B ∼= MALG) we conclude thatL∞(C) is isomorphic to C(X) ∼= C(St(RO(X)) ∼= C(St(MALG)).

Linear combinations of characteristic functions of measurable sets are dense in L∞(C). Thusany character k in the spectrum X of L∞(C) is univocally determined by its value on character-istic functions χAA∈MALG. In addition, since k ∈ X is a non zero homomorphism, given anyA ∈ MALG:

k(1) = k(χA + χAc) = 1

k(0) = k(χAχAc) = 0

This means that every characteristic function is mapped by k or in 0 or in 1, and if k(χA) = 1then k(χAc) = 0, and vice versa. Assume G ∈ St(MALG), then the function kG defined as

kG(χA) =

1 if A ∈ G0 if A /∈ G

and extended by linearity and continuity to L∞(C) belongs to the spectrum. Conversely, givenk ∈ X, we can consider the set Gk = A : k(χA) = 1 which can be easily verified to a ultrafilter.The maps

Θ : X → St(MALG)

k 7→ Gk


Ξ : St(MALG)→ X

G 7→ kG

are one the inverse of the other. If we show that one of them is continuous we obtain X ∼=St(MALG), since X and St(MALG) are both compact Hausdorff spaces. We will work on Θ.Consider OA a clopen set in St(MALG). Its preimage is the set of all k ∈ X such that k(χA) = 1,which is closed with respect to pointwise convergence, hence closed in X. In particular we getthat the preimage of OAc is open in X, thus we conclude that Θ is continuous. We can thereforewrite the isomorphism Λ (which is ΓL∞(C) composed with the isomorphism between C(X) andC(St(MALG))):

Λ : L∞(C)→ C(St(MALG))

f 7→ f

where f(G) = kG(f).

4.3 B-names for complex numbers

Through this section we will assume V to be a transitive model of ZFC and B ∈ V a booleanalgebra which V models to be complete. If G is a V -generic ultrafilter in B, V [G] will denote thegeneric extension of Definition 3.3.6.

Definition 4.3.1. σ ∈ V B is a B-name for a complex number if

Jσ is a complex numberK = 1B

We denote with CB the set of all B-names for complex numbers modulo the equivalence relation:

σ ≡ τ ⇔ Jσ = τK = 1B

Remark 4.3.2. Let

B = Un : n ∈ ω

be the family of the open balls in C whose radius and centre coordinates are rational. EveryBorel subset of C is obtained, in fewer than ℵ1 steps, from the elements of B by taking countableunions and complements. It is possible to code these operations with a real number r. Adetailed explanation of this procedure is given in section “Borel Codes” of [6, Chapter 25]. Forour purposes it is enough to say that if R is a Borel subset of C, there is a real number r ∈ Rand a (ZFC provably) ∆1-property P (x, y) such that

x ∈ R⇔ P (x, r)

Lemmas 25.24 and 25.25 in [6] show that these Borel codes and the inclusion relation betweenthese sets are absolute for transitive models. Consider two Borel sets R,S associated to r, s ∈ Rrespectively, and V a transitive model of ZFC such that r, s ∈ V . Let B be a complete booleanalgebra in V and assume there could be G a V -generic filter for B. The generic extension V [G]would be a transitive model of ZFC which contains V , hence the absoluteness properties betweenV and V [G] would guarantee that

RV ⊆ SV ⇔ RV [G] ⊆ SV [G]

4.3. B-names for complex numbers 55

whereRV = x ∈ V : P (x, r)

RV [G] = x ∈ V [G] : P (x, r)

and similarly for S. Guided by these considerations we define in V the following B-name:

R = (τ, JP (τ, r)K) : τ is a B-name for a complex number

It is easy to see that:RG = RV [G]

whenever G is a V -generic filter for B. In particular R ∈ V B is a canonical name to interpret theBorel relation R in any generic extension of V by a generic filter G.

We return now to CB. Observe that, given τ ∈ CB and a first order formula ϕ(x), the element

Jϕ(τ)KVB

is well-defined in B, even if τ is an equivalence class. This follows from Lemma 2.3.11.Keeping this in mind we give the following:

Definition 4.3.3. Given R a Borel n-ary relation on C we define, for σ1, . . . , σn ∈ CB:

JR(σ1, . . . , σn)KCB

=r

(σ1, . . . , σn) ∈ RzV B

and similarly for Borel functions.

Lemma 2.3.11 and Soundness Theorem 2.3.14 applied in V B guarantee that items i-vii inDefinition 2.3.1 hold for the structure

〈CB, RCB

1 , . . . , RCB

k , FCB

1 , . . . , FCB

l 〉

where each Ri (Fj) is an arbitrary Borel relation (function) on Cni (from Cmj to C). As aconsequence, CB is B-valued extension of C when we consider languages composed by Borelrelations and functions. We have seen that this is the case also for C(St(B)), and we will be ableto build an embedding of this space in CB. Unfortunately these two spaces are not isomorphic asB-valued models. In order to have an isomorphism we need to consider a larger set of functionsfrom St(B).

Definition 4.3.4. Let X be a compact Hausdorff extremely disconnected space. C+(X) is theset of continuous functions f from X to the one point compactification of the complex fieldC ∪ ∞ ∼= S2 such that f−1[∞] is nowhere dense.

We can define on C+(St(B)) a structure of B-valued extension of C as we did for C(St(B)) inExample 4.1.4. Our aim is the following:

Theorem 4.3.5. Fix a setL = Ri : i ∈ I ∪ Fj : j ∈ J

where:

• for i ∈ I, Ri is a Borel subset of Cni ;

• for j ∈ J , Fj is a Borel function from Cmj to C.

Then the B-valued models C+(St(B)) and CB in the language L (as defined in Example 4.1.4 andDefinition 4.3.3 respectively) are isomorphic.


With some more effort we will be able to show how the study of complex numbers in a genericextension can be reduced to the study of C(St(B)), and we will derive the following:

Theorem 4.3.6. Let V be a transitive model of ZFC, B ∈ V which V models to be a completeboolean algebra, and G a V -generic filter in B. Assume R1, . . . , Rs are ni-ary Borel relationsand f1, . . . , ft mj-ary Borel functions on C. Then

〈C, R1, . . . , Rs, f1, . . . , ft〉 ≺Σ2〈C(St(B))/G,R1/G, . . . , Rs/G, f1/G, . . . , ft/G〉

Before getting to these theorems, we need several intermediate results.Something similar to Remark 4.3.2 can be performed in St(B). In this case the parameters for

the “code” have to be taken among real numbers (to code the complexity of the Borel relation)and elements of B (to code the basic open sets), since a basis for St(B) is

Oa : a ∈ B

We will show explicitly the absoluteness of the inclusion relation for generic extensions in thecase of open and closed sets, since it will be needed later.

Lemma 4.3.7. Let V be a transitive models of ZFC, B ∈ V a boolean algebra which V modelsto be complete, and G a V -generic filter over B. Assume RV , SV are two open or closed sets inSt(B)V . Then

RV ⊆ SV ⇔ RV [G] ⊆ SV [G]

Proof. Clopen ⊆ Clopen: Given a, b ∈ B:

OVa ⊆ OVb ⇔ a ≤ b

The order relation in B is absolute for generic extensions, therefore the thesis follows.

Open ⊆ Closed: Let A,B ⊆ B. Then:⋃a∈AOVa ⊆

⋂b∈B

OVb ⇔ ∀a ∈ A∀b ∈ B(OVa ⊆ OVb )

⇔ ∀a ∈ A∀b ∈ B(OV [G]a ⊆ OV [G]

b )

⇔⋃a∈AOV [G]a ⊆

⋂b∈B

OV [G]b

Closed ⊆ Open: Given A,B ⊆ B, assume that we have⋂a∈AOVa ⊆

⋃b∈B

OVb

We define the set¬B = ¬b : b ∈ B

The hypothesis guarantees that A∪¬B does not have the finite intersection property. Sincethis property of A ∪ ¬B as a subset of B is absolute for generic extensions, this is true in

V [G] as well, therefore there is no H ∈⋂a∈AO

V [G]a such that

H ∈⋂b∈B

OV [G]¬b =

(⋃b∈B

OV [G]b

)c


Now assume that ⋂a∈AOVa 6⊆

⋃b∈B

OVb

Then the family A ∪ ¬B has the finite intersection property in V as well as in V [G], thuswe can find H ∈ St(B)V [G] such that

H ∈⋂a∈AOV [G]a ∩

⋂b∈B

OV [G]¬b

Open ⊆ Open: Given A,B ⊆ B, consider ⋃a∈AOVa ,

⋃b∈B

OVb

Since every OVa ∈ ∆01 ⊆ Π0

1, from previous items we have that:

OVa ⊆⋃b∈B

OVb ⇔ OV [G]a ⊆

⋃b∈B

OV [G]b

thus: ⋃a∈AOVa ⊆

⋃b∈B

OVb ⇔⋃a∈AOV [G]a ⊆

⋃b∈B

OV [G]b

Remark 4.3.8. In the following part, given B a complete boolean algebra, we will often confuseit with RO(St(B)). If U is a regular open set of St(B) and G ∈ St(B), we may write equivalently

G ∈ U,U ∈ G

depending if we are considering U as an element of RO(St(B)) or as the correspondent elementin B.

We recall the definition of B in Remark 4.3.2, the countable basis of C whose elements arethe open balls with rational radius and whose centre has rational coordinates:

B = Un : n ∈ ω

The following lemma characterizes the elements of C+(St(B)).

Lemma 4.3.9. Assume f ∈ V is an element of C+(St(B)). For H ∈ St(B) we define

ΣHf = Un :˚

f−1[Un] ∈ H

Then, for H ∈ St(B), we have:f(H) = σHf

where σHf it is the only element in⋂

ΣHf if ΣHf is non-empty, and σHf =∞ otherwise.

Proof. Assume ΣHf is empty. If f(H) ∈ Un for some n ∈ ω it follows that:

H ∈ f−1[Un] ⊆ ˚f−1[Un]

hence˚

f−1[Un] ∈ ΣHf , which is absurd. Suppose now that ΣHf is non-empty.


Claim 4.3.9.1. Assume ΣHf is non-empty. Then⋂

ΣHf is a singleton.

Proof. Let m ∈ ω be such that Um ∈ ΣHf .

Existence: The family

ΣHf = Um ∩ Un :˚

f−1[Un] ∈ H

is a family of closed subsets of Um. ΣHf inherits the finite intersection property from H,

hence so does ΣHf . We can conclude that

∅ 6=⋂

ΣHf ⊆⋂

ΣHf

Uniqueness: Suppose there are two different points x, y ∈⋂

ΣHf . There exists p ∈ ω such that

x ∈ Up, y /∈ Up. The last relation implies Up /∈ ΣHf . Now we show that if an element in⋂ΣHf is in a certain Un, then

˚f−1[Un] ∈ H. Therefore x ∈ Up implies Up ∈ ΣHf , which is

absurd. Suppose˚

f−1[Up] /∈ H, we have that:

H ∈ ¬ ˚f−1[Up] ∩ ˚

f−1[Um] ⊆ f−1[Um \ Up]

For each z ∈ Um \ Up there exists Unz such that

z ∈ Unz ∧ x /∈ Unz

This family of open balls covers the compact space Um \Up, so that there are z1, · · · , zk ∈Um \ Up which verify the following chain of inclusions:

f−1[Um \ Up] ⊆⋃

1≤i≤k

f−1[Unzi ] ⊆⋃

1≤i≤k

˚f−1[Unzi ]

There is therefore a zj such that˚

f−1[Unzj ] ∈ H, hence Uzj ∈ ΣHf . This is absurd since

x /∈ Uzj .

Suppose f(H) 6= σHf and consider two open balls U1, U2 in B such that

U1 ∩ U2 = ∅

f(H) ∈ U1

σHf ∈ U2

It easily follows that both˚

f−1[U1] and˚

f−1[U2] are in H (the second assertion, can be shownalong the same lines of the uniqueness proof in Claim 4.3.9.1). These two sets are disjoint, acontradiction follows.

Remark 4.3.10. Previous lemma shows that in ZFC, given f ∈ C+(St(B)), it holds

f(H) = x⇔ x = σHf


The latter is a (ZFC provably) ∆1-property with ω and an =˚

f−1[Un] : n ∈ N as parameters.Thus, given V a transitive model of ZFC, B a complete boolean algebra in V , G a V -generic filterin B, any f ∈ V element of C+(St(B))V can be extended in an absolute manner to V [G] by therule:

fV [G] : St(B)V [G] → CV [G]

H 7→ σHf

where σHf is defined as in the previous lemma through the set

ΣHf = Un : an ∈ H

Remark 4.3.11. The next proofs will often use Proposition 3.4.14 in order to show that for acertain formula ϕ and for some b ∈ B

JϕK ≥ b

In that proposition we work with M , a countable set which reflects enough axioms of ZFC aswell as the formula JϕK ≥ b (precisely M reflects Θϕ ∧ (JϕK ≥ b)), and with N , its transitiveimage through the Mostwoski’s Collapse. From now on, when using Proposition 3.4.14, we willwork under the assumption that there exists a countable M ≺ V with M ∈ V for V a model ofZFC containing the relevant parameters of the formula JϕK ≥ b. If the reader is disturbed by thisassumption (which is consistencywise stronger than ZFC), she/he can apply some compactnessarguments in combination with Proposition 3.4.14 to remove this assumption, keeping in mindthat in each of the following proofs we need to reflect only a finite number of properties of V ,thus we could always reduce our attention to a countable M ≺ Vα for a certain ordinal α.

Lemma 4.3.12. Fix V a transitive model of ZFC and B ∈ V a boolean algebra which V modelsto be complete. Let f ∈ C+(St(B)) and consider

B = Un : n ∈ ω

the countable basis of C defined in Remark 4.3.2. For each n ∈ ω let

an =˚

f−1[Un]

There exists a unique τf ∈ CB such that

rτf ∈ Un

z= an

Proof. Consider the B-nameΣf = (Un, an) : n ∈ ω

We start showing that r∃!x(x ∈

⋂Σf )

z= 1B

which, by Proposition 3.4.14, is a consequence of the following:

Claim 4.3.12.1. Assume M ≺ V is a countable model of ZFC such that ω ∪ an : n ∈ ω ∪B, f ⊆M , and π : M → N is the Mostowski’s Collapse. Let G be an N -generic filter for π(B).Then:

N [G] |= ∃!x(x ∈

⋂ΣGπ(f)

)where ΣGπ(f) = UN [G]

n : π(an) ∈ G.


Proof. Notice that since ω ⊆M is transitive, rational and complex numbers (the power-set of atransitive set is transitive) are preserved by π and that CN = C∩N . First, we prove that ΣGπ(f)

is non-empty (observe that π(f) preserves all the properties of f since π is an isomorphism).The preimage of CN through π(f) contains an open dense subset of St(π(B))N , hence (observethat π(f)−1[UNn ] = π(f−1[Un])) it follows that⋃

n∈ωπ(an)

is an open dense subset of St(π(B)) as well. By Lemma 3.3.4 we can infer that

D = a ∈ π(B)+ : ∃n ∈ ω(a ≤ π(an))

is open dense in π(B)+. Since G is N -generic, G ∩ D 6= ∅. There exists therefore m ∈ ω such

that π(am) ∈ G, thus UN [G]

m ∈ ΣGπ(f). The proof that⋂

ΣGπ(f) is a singleton can be carried in Nas in Claim 4.3.9.1.

V B is full, there is therefore a B-name τf such that

rτf ∈

⋂Σf

z= 1B

This is a B-name for a complex number. Moreover, if τ is a B-name for a complex number andrτ ∈

⋂Σf

z= 1B

then, from

(τf ∈⋂

Σf ) ∧ (τ ∈⋂

Σf ) ∧ (∃!x(x ∈⋂

Σf ))→ τ = τf

it follows that (by the Soundness Theorem 2.3.14):

Jτ = τf K = 1B

Now we focus on the proof of the thesis of the lemma. We will use again Proposition 3.4.14. LetM ≺ V be a countable structure as in Claim 4.3.12.1, π : M → N its Mostowski’s Collapse, andG an N -generic filter for π(B). On the one hand we have (using the same proof of the uniqueness

part in Claim 4.3.9.1) that if τGπ(f) ∈ UN [G]n then π(an) ∈ G, which means by Proposition 3.4.14

¬an ≤rτf /∈ Un

z

or equivalently rτf ∈ Un

z≤ an

On the other hand

G ∈ π(f)N [G]−1[UN [G]n ]⇒ π(f)N [G](G) = τGπ(f) ∈ U

N [G]n ⇒

rτπ(f) ∈ Un

zNπ(B)

∈ G

which means, interpretingrτπ(f) ∈ Un

zNπ(B)

as a clopen subset of St(π(B))N [G], that

π(f)N [G]−1[UN [G]n ] ⊆

(rτπ(f) ∈ Un

zNπ(B))N [G]


Lemmas 4.3.7 and 4.3.9 guarantee that this is equivalent to

π(f)−1[UNn ] ⊆rτπ(f) ∈ Un

zNπ(B)

Sincerτπ(f) ∈ Un

zNπ(B)

is clopen this implies that

π(an) ≤rτπ(f) ∈ Un

zNπ(B)

and therefore: (an ≤

rτf ∈ Un

z)Mwhich guarantees the thesis in V since M ≺ V .

Corollary 4.3.13. With the hypotheses of Lemma 4.3.12, if G is a V -generic filter in B then:

fV [G](G) = τGf

Moreover, fix R (F ) an n-ary Borel relation (function) over C and f1, . . . , fn+1 ∈ C+(St(B)).Then:

JR(τf1 , . . . , τfn)KCB

∈ G⇔ RV [G](fV [G]1 (G), . . . , fV [G]

n (G))

qF (τf1 . . . , τfn) = τfn+1

yCB

∈ G⇔ FV [G](fV [G]1 (G), . . . , fV [G]

n (G)) = fV [G]n+1 (G)

Proof. This follows from the previous result and from the definition of boolean interpretation ofBorel relations and functions in CB (Definition 4.3.3).

The map we have built in the lemma above is surjective.

Lemma 4.3.14. Assume τ ∈ CB. Consider

fτ : St(B)→ C ∪ ∞H 7→ σHf

where, given

ΣHf = Un :rτ ∈ Un

z∈ H

σHf is the only element in⋂

ΣHf if ΣHf is non-empty, σHf =∞ otherwise. The function f belongs

to C+(St(B)) and τfτ = τ .

Proof. The proof that if ΣHf is non-empty then its intersection has one single point can be carried

as in Claim 4.3.9.1 substituting˚

f−1[Un] withrτ ∈ Un

z.

Preimage of ∞ is nowhere dense: We show that the preimage of C through fτ contains anopen dense set. We use the following abbreviation

an =rτ ∈ Un

z

and consider the set A = an : n ∈ ω. First we show that:∨n∈ω

an = 1B


We use Proposition 3.4.14. Let M ≺ V be a countable structure such that B, τ ∈ M ,ω,A ⊆M , and as usual let π : M → N be the Mostowski’s Collapse. Since τ is a B-namefor a complex number in M , π(τ) is a π(B)-name for a complex number in N . Let G bean N -generic filter over π(B), we have therefore:

N [G] |= π(τ)G ∈ C

we can inferN [G] |= ∃n ∈ ω(π(τ)G ∈ Un)

Since generic extensions and π preserve ω, we get that ωG = ωN [G]. Thus, by Proposition3.4.14: ∨

n∈ωan =

r∃n ∈ ω(τ ∈ Un)

z≥ 1B

This means that A is predense because if b ∈ B+:

b = b ∧∨n∈ω

an =∨n∈ω

b ∧ an

so that 0B 6= ak ∧ b ≤ b for some k ∈ ω. We can conclude with Lemma 3.3.4.

Continuous: Let H ∈ St(B) be in the preimage of C, hence there exists m ∈ ω such that am ∈ Hand as a consequence

fτ (H) ∈ UmLet U be an open subset of C containing fτ (H), and consider Uk ∈ B such that

fτ (H) ∈ Uk

Uk ⊆ Um ∩ U

Since

fτ (H) ∈ Uk ⇒ ak ∈ H (1)

which can be showed as in the uniqueness part in Claim 4.3.9.1 substituting˚

f−1[Un] withrτ ∈ Un

z, and since the following inclusion holds:

Oak ⊆ f−1τ (U)

the continuity of fτ for points in the preimage of C is proved.

Consider now H ∈ f−1τ (∞). Let A be an open neighborhood of ∞, and let Uk be an

open ball centred in zero with rational radius such that

Uc

k ⊆ A

We also consider Ul such thatUk ⊆ Ul

By definition of fτ we have that H ∈ Ocal , and by (1) the image of any element in the openset Ocal can not belong to Ul. Thus

Ocal ⊆ f−1τ [U cl ] ⊆ f−1

τ [Uc

k] ⊆ f−1τ [A]


τfτ = τ : We already know that (see (1)):

f−1τ [Un] ⊆ Oan

The second set is clopen, thus:

rτfτ ∈ Un

z=

˚f−1τ [Un] ⊆ Oan (2)

Toward a contradiction, assume Jτ = τfτ K 6= 1B. By Proposition 3.4.14 we can find M ≺ Va countable structure with B, τ, f ∈M , ω ⊆M , and where π : M → N is the Mostowski’sCollapse, such that there is an N -generic filter G which verifies

N [G] |= π(τ)G 6= π(τfτ )G

Thus there exists n ∈ ω such that:

π(τfτ )G ∈ UN [G]n

π(τ)G /∈ UN [G]n

The inclusion relation (2) implies

rπ(τfτ ) ∈ Un

z≤

rπ(τ) ∈ Un

z

and by Cohen’s Forcing Theoremrπ(τfτ ) ∈ Un

z∈ G. This is a contradiction.

Finally we can show that the map f → τf is an isomorphism of B-valued models between CB

and C+(St(B)).

Theorem 4.3.15. Fix a set

L = Ri : i ∈ I ∪ Fj : j ∈ J

where:

• for i ∈ I, Ri is a Borel subset of Cni ;

• for j ∈ J , Fj is a Borel function from Cmj to C.

and consider C+(St(B)) and CB as B-valued models in the language L as defined in Example4.1.4 and Definition 4.3.3 respectively. The map

Ω : C+(St(B))→ CB

f 7→ τf

is an isomorphism of B-valued models.


Proof. We will first consider the case of R ⊆ C a unary Borel relation in C. Given f ∈ C+(St(B)),

consider JR(f)K andrτf ∈ R

zas regular open subsets of St(B). By Proposition 4.1.2, in order

to show that they overlap, it is sufficient to prove that their symmetric difference is meager. Bydefinition, we already know that JR(f)K has meager difference with the set

H ∈ St(B) : f(H) ∈ R = f−1[R]

Therefore it suffices to prove thatrτf ∈ R

zand f−1[R] have meager difference. The proof

proceeds step by step on the hierarchy of Borel sets Σ0α, Π0

α, for α a countable ordinal, as definedin [6, Chapter 11, Section 1].

Σ01: Let R be an element of the basis

B = Un : n ∈ ω

defined in Remark 4.3.2. The thesis follows from Lemma 4.3.12, in fact

rτf ∈ Un

z=

˚f−1[Un]

which has meager difference with f−1[Un]. Consider now

R =⋃i∈I

Ui

where I is a countable set of indexes. In this case we have that

f−1[R] =⋃i∈I

f−1[Ui]

and rτf ∈ R

z=∨i∈I

rτf ∈ Ui

z= A

where A =⋃i∈I

rτf ∈ Ui

z. For each i ∈ I, the sets f−1[Ui] and

rτf ∈ Ui

zhave meager

difference, thus f−1[R]∆A is meager. This is true since meager sets are closed for countableunion, and the following property holds:(⋃

k∈K

Bk

)∆

(⋃k∈K

Ck

)⊆⋃k∈K

(Bk∆Ck)

The proof is therefore concluded because A∆ A is meager.

Σ0α ⇒ Π0

α: Suppose R ∈ Π0α, and that the thesis holds for Borel sets in Σ0

α. By definition

Rc ∈ Σ0α, therefore:

f−1[Rc]∆rτf ∈ Rc

zis meager

SinceB∆C = Bc∆Cc

we have that

f−1[R]∆rτf ∈ Rc

zcis meager


Thus we can conclude, in fact

rτf ∈ R

z= ¬

rτf ∈ Rc

z

and the last set is the interior ofrτf ∈ Rc

zc, therefore they differ by a meager set.

Π0α ⇒ Σ0

α+1: This item can be proved as the second part of the case α = 1, substituting the Un

with Borel sets in Π0α.

Σ0β for β limit ordinal: If the thesis holds for α < β, then the proof can be carried similarly to

the case Π0α ⇒ Σ0

α+1.

This proves the theorem for unary Borel relations. The m-ary case (for both relation and functionsymbols) can be shown similarly starting from the open basis of Cm

Un1× · · · × Unm : n1, . . . , nm ∈ ω

Base step holds due to Lemma 4.3.12, and then we proceed by induction as we did earlier.Lemma 4.3.14 guarantees that Ω is surjective, thus the proof is concluded.

The morphism Ω had to be defined on C+(St(B)) in order to be surjective. Nevertheless,when passing to a generic extension of V , it is enough to consider the equivalence classes of theelement of the C∗-algebra C(St(B)).

Proposition 4.3.16. Assume V is a model of ZFC, B a complete boolean algebra in V and G aV -generic filter in B. Then

C+(St(B))/G ∼= C(St(B))/G

Proof. We need to show that for each f ∈ C+(St(B)) we can find a f ∈ C(St(B)) such that

rf = f

z∈ G

which, by Corollary 4.3.13, is equivalent to

fV [G](G) = fV [G](G)

We denote again:

an =˚

f−1[Un]

Proceeding as in Claim 4.3.12.1, we can find m ∈ ω such that am ∈ G. For each H ∈ Oam wehave that:

f(H) ∈ Umby Lemma 4.3.9. We can therefore consider the restriction of f to Oa and extend it to af ∈ C(St(B))1 The implication

f OVam= f OVam⇒ fV [G] OV [G]am

= fV [G] OV [G]am

guarantees the thesis since G ∈ OV [G]am .

1This is a fact of general topology, known as Tietze’s Extension Theorem. For a proof of the latter see [15,Theorem 15.8].


Remark 4.3.17. Once we fix a language of Borel relations and functions in C, the theoremsproved so far guarantee that the following two first order models are isomorphic whenever G isV -generic for B:

C(St(B))/G ∼= CB/G

If we consider for instance L = +, ∗, , we obtain that C(St(B))/G is an algebraically closedfield which extends C (modulo isomorphism).

We conclude combining the results of this section with Cohen’s Generic Absoluteness Theorem3.5.3.

Theorem 4.3.18. Let V be a transitive model of ZFC, B ∈ V which V models to be a completeboolean algebra, and G a ultrafilter on B. Assume R1, . . . , Rs are ni-ary Borel relations andf1, . . . , ft mj-ary Borel functions on C. Then

〈C, R1, . . . , Rs, f1, . . . , ft〉 ≺Σ2 〈C+(St(B))/G,R1/G, . . . , Rs/G, f1/G, . . . , ft/G〉

Moreover, if G is V -generic, then the following holds:

〈C, R1, . . . , Rs, f1, . . . , ft〉 ≺Σ2〈C(St(B))/G,R1/G, . . . , Rs/G, f1/G, . . . , ft/G〉

Proof. Cohen’s Generic Absoluteness Theorem 3.5.3 implies that

V ≺Σ1V B/G

Thus the thesis is a consequence of

C+(St(B))/G ∼= CB/G

and of Lemma 25.25 in [6], which provides a translation of Σ12-formulae to Σ1-formulae in V .

The second part follows from Proposition 4.3.16.

This is the last theorem we present. The results of this chapter might be a starting point anduseful tools in order to prove properties of the complex field using ideas arising from the theoryof C∗-algebras, and vice versa. Applications of these results are topic of present research.

69

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